🕷️ Crawler Inspector

URL Lookup

Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 184 (from laksa003)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa184.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://mathworld.wolfram.com/Eigenvalue.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://mathworld.wolfram.com/Eigenvalue.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/mathworld.wolfram.com\/Eigenvalue.html","crawl_time":1776415169,"first_indexed_time":1588012248,"http_code":200,"src_unparsed":"com,wolfram!mathworld,\/Eigenvalue.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"Eigenvalue -- from Wolfram MathWorld","meta_descriptions":["Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability..."],"attrs_boilerpipe_text":"Algebra\nApplied Mathematics\nCalculus and Analysis\nDiscrete Mathematics\nFoundations of Mathematics\nGeometry\nHistory and Terminology\nNumber Theory\nProbability and Statistics\nRecreational Mathematics\nTopology\nAlphabetical Index\nNew in MathWorld\nEigenvalues are a special set of scalars associated with a\nlinear system of equations\n(i.e., a\nmatrix equation\n)\n that are sometimes also known as characteristic roots, characteristic values (Hoffman\n and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).\nThe determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to\nmatrix\n diagonalization\nand arises in such common applications as stability analysis,\n the physics of rotating bodies, and small oscillations of vibrating systems, to name\n only a few. Each eigenvalue is paired with a corresponding so-called\neigenvector\n(or, in general, a corresponding\nright eigenvector\nand a corresponding\nleft eigenvector\n; there is\n no analogous distinction between left and right for eigenvalues).\nThe decomposition of a\nsquare matrix\ninto eigenvalues and eigenvectors is known in this work as\neigen decomposition\n, and the fact that this\n decomposition is always possible as long as the matrix consisting of the eigenvectors\n of\nis\nsquare\nis known as the\neigen decomposition theorem\n.\nThe\nLanczos algorithm\nis an algorithm for computing the eigenvalues and\neigenvectors\nfor large\nsymmetric\nsparse matrices\n.\nLet\nbe a\nlinear\n transformation\nrepresented by a\nmatrix\n. If there is a\nvector\nsuch that\n(1)\nfor some\nscalar\n, then\nis called the eigenvalue of\nwith corresponding (right)\neigenvector\n.\nLetting\nbe a\nsquare\n matrix\n(2)\nwith eigenvalue\n,\n then the corresponding\neigenvectors\nsatisfy\n(3)\nwhich is equivalent to the homogeneous system\n(4)\nEquation (\n4\n) can be written compactly as\n(5)\nwhere\nis the\nidentity matrix\n. As shown in\nCramer's\n rule\n, a\nlinear system of equations\nhas nontrivial solutions\niff\nthe\ndeterminant\nvanishes, so the solutions of equation (\n5\n) are given by\n(6)\nThis equation is known as the\ncharacteristic equation\nof\n,\n and the left-hand side is known as the\ncharacteristic\n polynomial\n.\nFor example, for a\nmatrix, the eigenvalues are\n(7)\nwhich arises as the solutions of the\ncharacteristic\nequation\n(8)\nIf all\neigenvalues are different, then plugging these back in gives\nindependent equations for the\ncomponents of each corresponding\neigenvector\n,\n and the system is said to be nondegenerate. If the eigenvalues are\n-fold\ndegenerate\n, then the system\n is said to be degenerate and the\neigenvectors\nare\n not linearly independent. In such cases, the additional constraint that the\neigenvectors\nbe\northogonal\n,\n(9)\nwhere\nis the\nKronecker delta\n, can be applied to yield\nadditional constraints, thus allowing\n solution for the\neigenvectors\n.\nEigenvalues may be computed in the\nWolfram Language\nusing\nEigenvalues\n[\nmatrix\n].\n Eigenvectors and eigenvalues can be returned together using the command\nEigensystem\n[\nmatrix\n].\nAssume we know the eigenvalue for\n(10)\nAdding a constant times the\nidentity matrix\nto\n,\n(11)\nso the new eigenvalues equal the old plus\n. Multiplying\nby a constant\n(12)\nso the new eigenvalues are the old multiplied by\n.\nNow consider a\nsimilarity transformation\nof\n. Let\nbe the\ndeterminant\nof\n, then\n(13)\n(14)\n(15)\nso the eigenvalues are the same as for\n.\nSee also\nBrauer's Theorem\n,\nCharacteristic Equation\n,\nCharacteristic Polynomial\n,\nComplex Matrix\n,\nCondition\n Number\n,\nEigen Decomposition\n,\nEigen\n Decomposition Theorem\n,\nEigenfunction\n,\nEigenvector\n,\nFrobenius Theorem\n,\nGershgorin\n Circle Theorem\n,\nLanczos Algorithm\n,\nLyapunov's\n First Theorem\n,\nLyapunov's Second Theorem\n,\nMatrix Diagonalization\n,\nOstrowski's\n Theorem\n,\nPerron's Theorem\n,\nPerron-Frobenius\n Theorem\n,\nPoincaré Separation\n Theorem\n,\nRandom Matrix\n,\nReal\n Matrix\n,\nSchur's Inequalities\n,\nSimilarity\n Transformation\n,\nSturmian Separation\n Theorem\n,\nSylvester's Inertia Law\n,\nWielandt's Theorem\nExplore this topic in the MathWorld classroom\nExplore with Wolfram|Alpha\nReferences\nArfken, G. \"Eigenvectors, Eigenvalues.\" §4.7 in\nMathematical\n Methods for Physicists, 3rd ed.\nOrlando, FL: Academic Press, pp. 229-237,\n 1985.\nHoffman, K. and Kunze, R. \"Characteristic Values.\" §6.2\n in\nLinear\n Algebra, 2nd ed.\nEnglewood Cliffs, NJ: Prentice-Hall, p. 182, 1971.\nKaltofen,\n E. \"Challenges of Symbolic Computation: My Favorite Open Problems.\"\nJ.\n Symb. Comput.\n29\n, 891-919, 2000.\nMarcus, M. and Minc, H.\nIntroduction\n to Linear Algebra.\nNew York: Dover, p. 145, 1988.\nNash,\n J. C. \"The Algebraic Eigenvalue Problem.\" Ch. 9 in\nCompact\n Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.\nBristol, England: Adam Hilger, pp. 102-118, 1990.\nPress, W. H.;\n Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. \"Eigensystems.\"\n Ch. 11 in\nNumerical\n Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.\nCambridge, England:\n Cambridge University Press, pp. 449-489, 1992.\nReferenced on Wolfram|Alpha\nEigenvalue\nCite this as:\nWeisstein, Eric W.\n\"Eigenvalue.\" From\nMathWorld\n--A Wolfram Resource.\nhttps:\/\/mathworld.wolfram.com\/Eigenvalue.html\nSubject classifications","attrs_markdown":"![](https:\/\/mathworld.wolfram.com\/images\/header\/menu-icon.png) TOPICS\n\n![](https:\/\/mathworld.wolfram.com\/images\/sidebar\/menu-close.png)\n\n[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n- [Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)\n- [Linear Algebra](https:\/\/mathworld.wolfram.com\/topics\/LinearAlgebra.html)\n- [Matrices](https:\/\/mathworld.wolfram.com\/topics\/Matrices.html)\n- [Matrix Eigenvalues](https:\/\/mathworld.wolfram.com\/topics\/MatrixEigenvalues.html)\n\n- [Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)\n- [Linear Algebra](https:\/\/mathworld.wolfram.com\/topics\/LinearAlgebra.html)\n- [Matrices](https:\/\/mathworld.wolfram.com\/topics\/Matrices.html)\n- [Matrix Decomposition](https:\/\/mathworld.wolfram.com\/topics\/MatrixDecomposition.html)\n# Eigenvalue\n***\n[![DOWNLOAD Mathematica Notebook](https:\/\/mathworld.wolfram.com\/images\/entries\/download-notebook-icon.png)Download Wolfram Notebook](https:\/\/mathworld.wolfram.com\/notebooks\/LinearAlgebra\/Eigenvalue.nb)\n\nEigenvalues are a special set of scalars associated with a [linear system of equations](https:\/\/mathworld.wolfram.com\/LinearSystemofEquations.html) (i.e., a [matrix equation](https:\/\/mathworld.wolfram.com\/MatrixEquation.html)) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).\n\nThe determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to [matrix diagonalization](https:\/\/mathworld.wolfram.com\/MatrixDiagonalization.html) and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html) (or, in general, a corresponding [right eigenvector](https:\/\/mathworld.wolfram.com\/RightEigenvector.html) and a corresponding [left eigenvector](https:\/\/mathworld.wolfram.com\/LeftEigenvector.html); there is no analogous distinction between left and right for eigenvalues).\n\nThe decomposition of a [square matrix](https:\/\/mathworld.wolfram.com\/SquareMatrix.html) ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline1.svg) into eigenvalues and eigenvectors is known in this work as [eigen decomposition](https:\/\/mathworld.wolfram.com\/EigenDecomposition.html), and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline2.svg) is [square](https:\/\/mathworld.wolfram.com\/SquareMatrix.html) is known as the [eigen decomposition theorem](https:\/\/mathworld.wolfram.com\/EigenDecompositionTheorem.html).\n\nThe [Lanczos algorithm](https:\/\/mathworld.wolfram.com\/LanczosAlgorithm.html) is an algorithm for computing the eigenvalues and [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) for large [symmetric](https:\/\/mathworld.wolfram.com\/SymmetricMatrix.html) [sparse matrices](https:\/\/mathworld.wolfram.com\/SparseMatrix.html).\n\nLet ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline3.svg) be a [linear transformation](https:\/\/mathworld.wolfram.com\/LinearTransformation.html) represented by a [matrix](https:\/\/mathworld.wolfram.com\/Matrix.html) ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline4.svg). If there is a [vector](https:\/\/mathworld.wolfram.com\/Vector.html) ![X in R^n!=0](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline5.svg) such that\n\n|   |   |\n|---|---|\n| ![ AX=lambdaX ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation1.svg) | (1) |\n\nfor some [scalar](https:\/\/mathworld.wolfram.com\/Scalar.html) ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline6.svg), then ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline7.svg) is called the eigenvalue of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline8.svg) with corresponding (right) [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html) ![X](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline9.svg).\n\nLetting ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline10.svg) be a ![k×k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline11.svg) [square matrix](https:\/\/mathworld.wolfram.com\/SquareMatrix.html)\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11) a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22) ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)\\] ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation2.svg) | (2) |\n\nwith eigenvalue ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline12.svg), then the corresponding [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) satisfy\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11) a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22) ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)\\]\\[x\\_1; x\\_2; \\|; x\\_k\\]=lambda\\[x\\_1; x\\_2; \\|; x\\_k\\], ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation3.svg) | (3) |\n\nwhich is equivalent to the homogeneous system\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11)-lambda a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22)-lambda ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)-lambda\\]\\[x\\_1; x\\_2; \\|; x\\_k\\]=\\[0; 0; \\|; 0\\]. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation4.svg) | (4) |\n\nEquation ([4](https:\/\/mathworld.wolfram.com\/Eigenvalue.html#eqn4)) can be written compactly as\n\n|   |   |\n|---|---|\n| ![ (A-lambdaI)X=0, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation5.svg) | (5) |\n\nwhere ![I](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline13.svg) is the [identity matrix](https:\/\/mathworld.wolfram.com\/IdentityMatrix.html). As shown in [Cramer's rule](https:\/\/mathworld.wolfram.com\/CramersRule.html), a [linear system of equations](https:\/\/mathworld.wolfram.com\/LinearSystemofEquations.html) has nontrivial solutions [iff](https:\/\/mathworld.wolfram.com\/Iff.html) the [determinant](https:\/\/mathworld.wolfram.com\/Determinant.html) vanishes, so the solutions of equation ([5](https:\/\/mathworld.wolfram.com\/Eigenvalue.html#eqn5)) are given by\n\n|   |   |\n|---|---|\n| ![ det(A-lambdaI)=0. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation6.svg) | (6) |\n\nThis equation is known as the [characteristic equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline14.svg), and the left-hand side is known as the [characteristic polynomial](https:\/\/mathworld.wolfram.com\/CharacteristicPolynomial.html).\n\nFor example, for a ![2×2](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline15.svg) matrix, the eigenvalues are\n\n|   |   |\n|---|---|\n| ![ lambda\\_+\/-=1\/2\\[(a\\_(11)+a\\_(22))+\/-sqrt(4a\\_(12)a\\_(21)+(a\\_(11)-a\\_(22))^2)\\], ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation7.svg) | (7) |\n\nwhich arises as the solutions of the [characteristic equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html)\n\n|   |   |\n|---|---|\n| ![ x^2-x(a\\_(11)+a\\_(22))+(a\\_(11)a\\_(22)-a\\_(12)a\\_(21))=0. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation8.svg) | (8) |\n\nIf all ![k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline16.svg) eigenvalues are different, then plugging these back in gives ![k-1](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline17.svg) independent equations for the ![k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline18.svg) components of each corresponding [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html), and the system is said to be nondegenerate. If the eigenvalues are ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline19.svg)\\-fold [degenerate](https:\/\/mathworld.wolfram.com\/Degenerate.html), then the system is said to be degenerate and the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) are not linearly independent. In such cases, the additional constraint that the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) be [orthogonal](https:\/\/mathworld.wolfram.com\/OrthogonalVectors.html),\n\n|   |   |\n|---|---|\n| ![ X\\_i·X\\_j=\\|X\\_i\\|\\|X\\_j\\|delta\\_(ij), ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation9.svg) | (9) |\n\nwhere ![delta\\_(ij)](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline20.svg) is the [Kronecker delta](https:\/\/mathworld.wolfram.com\/KroneckerDelta.html), can be applied to yield ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline21.svg) additional constraints, thus allowing solution for the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html).\n\nEigenvalues may be computed in the [Wolfram Language](http:\/\/www.wolfram.com\/language\/) using [Eigenvalues](http:\/\/reference.wolfram.com\/language\/ref\/Eigenvalues.html)\\[*matrix*\\]. Eigenvectors and eigenvalues can be returned together using the command [Eigensystem](http:\/\/reference.wolfram.com\/language\/ref\/Eigensystem.html)\\[*matrix*\\].\n\nAssume we know the eigenvalue for\n\n|   |   |\n|---|---|\n| ![ AX=lambdaX. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation10.svg) | (10) |\n\nAdding a constant times the [identity matrix](https:\/\/mathworld.wolfram.com\/IdentityMatrix.html) to ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline22.svg),\n\n|   |   |\n|---|---|\n| ![ (A+cI)X=(lambda+c)X=lambda^'X, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation11.svg) | (11) |\n\nso the new eigenvalues equal the old plus ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline23.svg). Multiplying ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline24.svg) by a constant ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline25.svg)\n\n|   |   |\n|---|---|\n| ![ (cA)X=c(lambdaX)=lambda^'X, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation12.svg) | (12) |\n\nso the new eigenvalues are the old multiplied by ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline26.svg).\n\nNow consider a [similarity transformation](https:\/\/mathworld.wolfram.com\/SimilarityTransformation.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline27.svg). Let ![\\|A\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline28.svg) be the [determinant](https:\/\/mathworld.wolfram.com\/Determinant.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline29.svg), then\n\n|   |   |   |   |\n|---|---|---|---|\n| ![\\|Z^(-1)AZ-lambdaI\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline30.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline31.svg) | ![\\|Z^(-1)(A-lambdaI)Z\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline32.svg) | (13) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline33.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline34.svg) | ![\\|Z\\|\\|A-lambdaI\\|\\|Z^(-1)\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline35.svg) | (14) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline36.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline37.svg) | ![\\|A-lambdaI\\|,](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline38.svg) | (15) |\n\nso the eigenvalues are the same as for ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline39.svg).\n***\n## See also\n[Brauer's Theorem](https:\/\/mathworld.wolfram.com\/BrauersTheorem.html), [Characteristic Equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html), [Characteristic Polynomial](https:\/\/mathworld.wolfram.com\/CharacteristicPolynomial.html), [Complex Matrix](https:\/\/mathworld.wolfram.com\/ComplexMatrix.html), [Condition Number](https:\/\/mathworld.wolfram.com\/ConditionNumber.html), [Eigen Decomposition](https:\/\/mathworld.wolfram.com\/EigenDecomposition.html), [Eigen Decomposition Theorem](https:\/\/mathworld.wolfram.com\/EigenDecompositionTheorem.html), [Eigenfunction](https:\/\/mathworld.wolfram.com\/Eigenfunction.html), [Eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html), [Frobenius Theorem](https:\/\/mathworld.wolfram.com\/FrobeniusTheorem.html), [Gershgorin Circle Theorem](https:\/\/mathworld.wolfram.com\/GershgorinCircleTheorem.html), [Lanczos Algorithm](https:\/\/mathworld.wolfram.com\/LanczosAlgorithm.html), [Lyapunov's First Theorem](https:\/\/mathworld.wolfram.com\/LyapunovsFirstTheorem.html), [Lyapunov's Second Theorem](https:\/\/mathworld.wolfram.com\/LyapunovsSecondTheorem.html), [Matrix Diagonalization](https:\/\/mathworld.wolfram.com\/MatrixDiagonalization.html), [Ostrowski's Theorem](https:\/\/mathworld.wolfram.com\/OstrowskisTheorem.html), [Perron's Theorem](https:\/\/mathworld.wolfram.com\/PerronsTheorem.html), [Perron-Frobenius Theorem](https:\/\/mathworld.wolfram.com\/Perron-FrobeniusTheorem.html), [Poincaré Separation Theorem](https:\/\/mathworld.wolfram.com\/PoincareSeparationTheorem.html), [Random Matrix](https:\/\/mathworld.wolfram.com\/RandomMatrix.html), [Real Matrix](https:\/\/mathworld.wolfram.com\/RealMatrix.html), [Schur's Inequalities](https:\/\/mathworld.wolfram.com\/SchursInequalities.html), [Similarity Transformation](https:\/\/mathworld.wolfram.com\/SimilarityTransformation.html), [Sturmian Separation Theorem](https:\/\/mathworld.wolfram.com\/SturmianSeparationTheorem.html), [Sylvester's Inertia Law](https:\/\/mathworld.wolfram.com\/SylvestersInertiaLaw.html), [Wielandt's Theorem](https:\/\/mathworld.wolfram.com\/WielandtsTheorem.html) [Explore this topic in the MathWorld classroom](https:\/\/mathworld.wolfram.com\/classroom\/Eigenvalue.html)\n\n## Explore with Wolfram\\|Alpha\n![WolframAlpha](https:\/\/mathworld.wolfram.com\/images\/wolframalpha\/WA-logo.png)\n\nMore things to try:\n\n- [eigenvalue calculator](https:\/\/www.wolframalpha.com\/input\/?i=eigenvalue+calculator)\n- [eigenvalue](https:\/\/www.wolframalpha.com\/input\/?i=eigenvalue)\n- [eigenvalue {{0,1},{1,0}}](https:\/\/www.wolframalpha.com\/input\/?i=eigenvalue+{{0%2C1}%2C{1%2C0}})\n## References\nArfken, G. \"Eigenvectors, Eigenvalues.\" §4.7 in *[Mathematical Methods for Physicists, 3rd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0120598760\/ref=nosim\/ericstreasuretro)* Orlando, FL: Academic Press, pp. 229-237, 1985.\n\nHoffman, K. and Kunze, R. \"Characteristic Values.\" §6.2 in *[Linear Algebra, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0135367972\/ref=nosim\/ericstreasuretro)* Englewood Cliffs, NJ: Prentice-Hall, p. 182, 1971.\n\nKaltofen, E. \"Challenges of Symbolic Computation: My Favorite Open Problems.\" *J. Symb. Comput.* **29**, 891-919, 2000.\n\nMarcus, M. and Minc, H. *[Introduction to Linear Algebra.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0486656950\/ref=nosim\/ericstreasuretro)* New York: Dover, p. 145, 1988.\n\nNash, J. C. \"The Algebraic Eigenvalue Problem.\" Ch. 9 in *[Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/085274319X\/ref=nosim\/ericstreasuretro)* Bristol, England: Adam Hilger, pp. 102-118, 1990.\n\nPress, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. \"Eigensystems.\" Ch. 11 in *[Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/052143064X\/ref=nosim\/ericstreasuretro)* Cambridge, England: Cambridge University Press, pp. 449-489, 1992.\n\n## Referenced on Wolfram\\|Alpha\n[Eigenvalue](https:\/\/www.wolframalpha.com\/input\/?i=eigenvalue \"Eigenvalue\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Eigenvalue.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/Eigenvalue.html>\n\n## Subject classifications\n- [Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)\n- [Linear Algebra](https:\/\/mathworld.wolfram.com\/topics\/LinearAlgebra.html)\n- [Matrices](https:\/\/mathworld.wolfram.com\/topics\/Matrices.html)\n- [Matrix Eigenvalues](https:\/\/mathworld.wolfram.com\/topics\/MatrixEigenvalues.html)\n\n- [Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)\n- [Linear Algebra](https:\/\/mathworld.wolfram.com\/topics\/LinearAlgebra.html)\n- [Matrices](https:\/\/mathworld.wolfram.com\/topics\/Matrices.html)\n- [Matrix Decomposition](https:\/\/mathworld.wolfram.com\/topics\/MatrixDecomposition.html)\n\n- [About MathWorld](https:\/\/mathworld.wolfram.com\/about\/)\n- [MathWorld Classroom](https:\/\/mathworld.wolfram.com\/classroom\/)\n- [Contribute](https:\/\/mathworld.wolfram.com\/contact\/)\n- [MathWorld Book](https:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1420072218\/ref=nosim\/weisstein-20)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n\n- [13,311 Entries](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [Last Updated: Wed Mar 25 2026](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [©1999–2026 Wolfram Research, Inc.](https:\/\/www.wolfram.com\/)\n- [Terms of Use](https:\/\/www.wolfram.com\/legal\/terms\/mathworld.html)\n\n- [![Wolfram](https:\/\/mathworld.wolfram.com\/images\/footer\/wolfram-logo.png)](https:\/\/www.wolfram.com\/)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education\/)\n- Created, developed and nurtured by Eric Weisstein at Wolfram Research\n\n*Created, developed and nurtured by Eric Weisstein at Wolfram Research*\n\n[Find out if you already have access to Wolfram tech through your organization](https:\/\/www.wolfram.com\/siteinfo\/)\n\n×","attrs_readable_markdown":"[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n***\nEigenvalues are a special set of scalars associated with a [linear system of equations](https:\/\/mathworld.wolfram.com\/LinearSystemofEquations.html) (i.e., a [matrix equation](https:\/\/mathworld.wolfram.com\/MatrixEquation.html)) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).\n\nThe determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to [matrix diagonalization](https:\/\/mathworld.wolfram.com\/MatrixDiagonalization.html) and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html) (or, in general, a corresponding [right eigenvector](https:\/\/mathworld.wolfram.com\/RightEigenvector.html) and a corresponding [left eigenvector](https:\/\/mathworld.wolfram.com\/LeftEigenvector.html); there is no analogous distinction between left and right for eigenvalues).\n\nThe decomposition of a [square matrix](https:\/\/mathworld.wolfram.com\/SquareMatrix.html) ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline1.svg) into eigenvalues and eigenvectors is known in this work as [eigen decomposition](https:\/\/mathworld.wolfram.com\/EigenDecomposition.html), and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline2.svg) is [square](https:\/\/mathworld.wolfram.com\/SquareMatrix.html) is known as the [eigen decomposition theorem](https:\/\/mathworld.wolfram.com\/EigenDecompositionTheorem.html).\n\nThe [Lanczos algorithm](https:\/\/mathworld.wolfram.com\/LanczosAlgorithm.html) is an algorithm for computing the eigenvalues and [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) for large [symmetric](https:\/\/mathworld.wolfram.com\/SymmetricMatrix.html) [sparse matrices](https:\/\/mathworld.wolfram.com\/SparseMatrix.html).\n\nLet ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline3.svg) be a [linear transformation](https:\/\/mathworld.wolfram.com\/LinearTransformation.html) represented by a [matrix](https:\/\/mathworld.wolfram.com\/Matrix.html) ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline4.svg). If there is a [vector](https:\/\/mathworld.wolfram.com\/Vector.html) ![X in R^n!=0](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline5.svg) such that\n\n|   |   |\n|---|---|\n| ![ AX=lambdaX ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation1.svg) | (1) |\n\nfor some [scalar](https:\/\/mathworld.wolfram.com\/Scalar.html) ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline6.svg), then ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline7.svg) is called the eigenvalue of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline8.svg) with corresponding (right) [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html) ![X](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline9.svg).\n\nLetting ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline10.svg) be a ![k×k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline11.svg) [square matrix](https:\/\/mathworld.wolfram.com\/SquareMatrix.html)\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11) a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22) ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)\\] ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation2.svg) | (2) |\n\nwith eigenvalue ![lambda](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline12.svg), then the corresponding [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) satisfy\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11) a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22) ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)\\]\\[x\\_1; x\\_2; \\|; x\\_k\\]=lambda\\[x\\_1; x\\_2; \\|; x\\_k\\], ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation3.svg) | (3) |\n\nwhich is equivalent to the homogeneous system\n\n|   |   |\n|---|---|\n| ![ \\[a\\_(11)-lambda a\\_(12) ... a\\_(1k); a\\_(21) a\\_(22)-lambda ... a\\_(2k); \\| \\| ... \\|; a\\_(k1) a\\_(k2) ... a\\_(kk)-lambda\\]\\[x\\_1; x\\_2; \\|; x\\_k\\]=\\[0; 0; \\|; 0\\]. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation4.svg) | (4) |\n\nEquation ([4](https:\/\/mathworld.wolfram.com\/Eigenvalue.html#eqn4)) can be written compactly as\n\n|   |   |\n|---|---|\n| ![ (A-lambdaI)X=0, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation5.svg) | (5) |\n\nwhere ![I](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline13.svg) is the [identity matrix](https:\/\/mathworld.wolfram.com\/IdentityMatrix.html). As shown in [Cramer's rule](https:\/\/mathworld.wolfram.com\/CramersRule.html), a [linear system of equations](https:\/\/mathworld.wolfram.com\/LinearSystemofEquations.html) has nontrivial solutions [iff](https:\/\/mathworld.wolfram.com\/Iff.html) the [determinant](https:\/\/mathworld.wolfram.com\/Determinant.html) vanishes, so the solutions of equation ([5](https:\/\/mathworld.wolfram.com\/Eigenvalue.html#eqn5)) are given by\n\n|   |   |\n|---|---|\n| ![ det(A-lambdaI)=0. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation6.svg) | (6) |\n\nThis equation is known as the [characteristic equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline14.svg), and the left-hand side is known as the [characteristic polynomial](https:\/\/mathworld.wolfram.com\/CharacteristicPolynomial.html).\n\nFor example, for a ![2×2](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline15.svg) matrix, the eigenvalues are\n\n|   |   |\n|---|---|\n| ![ lambda\\_+\/-=1\/2\\[(a\\_(11)+a\\_(22))+\/-sqrt(4a\\_(12)a\\_(21)+(a\\_(11)-a\\_(22))^2)\\], ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation7.svg) | (7) |\n\nwhich arises as the solutions of the [characteristic equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html)\n\n|   |   |\n|---|---|\n| ![ x^2-x(a\\_(11)+a\\_(22))+(a\\_(11)a\\_(22)-a\\_(12)a\\_(21))=0. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation8.svg) | (8) |\n\nIf all ![k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline16.svg) eigenvalues are different, then plugging these back in gives ![k-1](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline17.svg) independent equations for the ![k](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline18.svg) components of each corresponding [eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html), and the system is said to be nondegenerate. If the eigenvalues are ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline19.svg)\\-fold [degenerate](https:\/\/mathworld.wolfram.com\/Degenerate.html), then the system is said to be degenerate and the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) are not linearly independent. In such cases, the additional constraint that the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html) be [orthogonal](https:\/\/mathworld.wolfram.com\/OrthogonalVectors.html),\n\n|   |   |\n|---|---|\n| ![ X\\_i·X\\_j=\\|X\\_i\\|\\|X\\_j\\|delta\\_(ij), ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation9.svg) | (9) |\n\nwhere ![delta\\_(ij)](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline20.svg) is the [Kronecker delta](https:\/\/mathworld.wolfram.com\/KroneckerDelta.html), can be applied to yield ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline21.svg) additional constraints, thus allowing solution for the [eigenvectors](https:\/\/mathworld.wolfram.com\/Eigenvector.html).\n\nEigenvalues may be computed in the [Wolfram Language](http:\/\/www.wolfram.com\/language\/) using [Eigenvalues](http:\/\/reference.wolfram.com\/language\/ref\/Eigenvalues.html)\\[*matrix*\\]. Eigenvectors and eigenvalues can be returned together using the command [Eigensystem](http:\/\/reference.wolfram.com\/language\/ref\/Eigensystem.html)\\[*matrix*\\].\n\nAssume we know the eigenvalue for\n\n|   |   |\n|---|---|\n| ![ AX=lambdaX. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation10.svg) | (10) |\n\nAdding a constant times the [identity matrix](https:\/\/mathworld.wolfram.com\/IdentityMatrix.html) to ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline22.svg),\n\n|   |   |\n|---|---|\n| ![ (A+cI)X=(lambda+c)X=lambda^'X, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation11.svg) | (11) |\n\nso the new eigenvalues equal the old plus ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline23.svg). Multiplying ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline24.svg) by a constant ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline25.svg)\n\n|   |   |\n|---|---|\n| ![ (cA)X=c(lambdaX)=lambda^'X, ](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/NumberedEquation12.svg) | (12) |\n\nso the new eigenvalues are the old multiplied by ![c](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline26.svg).\n\nNow consider a [similarity transformation](https:\/\/mathworld.wolfram.com\/SimilarityTransformation.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline27.svg). Let ![\\|A\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline28.svg) be the [determinant](https:\/\/mathworld.wolfram.com\/Determinant.html) of ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline29.svg), then\n\n|   |   |   |   |\n|---|---|---|---|\n| ![\\|Z^(-1)AZ-lambdaI\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline30.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline31.svg) | ![\\|Z^(-1)(A-lambdaI)Z\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline32.svg) | (13) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline33.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline34.svg) | ![\\|Z\\|\\|A-lambdaI\\|\\|Z^(-1)\\|](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline35.svg) | (14) |\n| ![](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline36.svg) | ![\\=](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline37.svg) | ![\\|A-lambdaI\\|,](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline38.svg) | (15) |\n\nso the eigenvalues are the same as for ![A](https:\/\/mathworld.wolfram.com\/images\/equations\/Eigenvalue\/Inline39.svg).\n***\n## See also\n[Brauer's Theorem](https:\/\/mathworld.wolfram.com\/BrauersTheorem.html), [Characteristic Equation](https:\/\/mathworld.wolfram.com\/CharacteristicEquation.html), [Characteristic Polynomial](https:\/\/mathworld.wolfram.com\/CharacteristicPolynomial.html), [Complex Matrix](https:\/\/mathworld.wolfram.com\/ComplexMatrix.html), [Condition Number](https:\/\/mathworld.wolfram.com\/ConditionNumber.html), [Eigen Decomposition](https:\/\/mathworld.wolfram.com\/EigenDecomposition.html), [Eigen Decomposition Theorem](https:\/\/mathworld.wolfram.com\/EigenDecompositionTheorem.html), [Eigenfunction](https:\/\/mathworld.wolfram.com\/Eigenfunction.html), [Eigenvector](https:\/\/mathworld.wolfram.com\/Eigenvector.html), [Frobenius Theorem](https:\/\/mathworld.wolfram.com\/FrobeniusTheorem.html), [Gershgorin Circle Theorem](https:\/\/mathworld.wolfram.com\/GershgorinCircleTheorem.html), [Lanczos Algorithm](https:\/\/mathworld.wolfram.com\/LanczosAlgorithm.html), [Lyapunov's First Theorem](https:\/\/mathworld.wolfram.com\/LyapunovsFirstTheorem.html), [Lyapunov's Second Theorem](https:\/\/mathworld.wolfram.com\/LyapunovsSecondTheorem.html), [Matrix Diagonalization](https:\/\/mathworld.wolfram.com\/MatrixDiagonalization.html), [Ostrowski's Theorem](https:\/\/mathworld.wolfram.com\/OstrowskisTheorem.html), [Perron's Theorem](https:\/\/mathworld.wolfram.com\/PerronsTheorem.html), [Perron-Frobenius Theorem](https:\/\/mathworld.wolfram.com\/Perron-FrobeniusTheorem.html), [Poincaré Separation Theorem](https:\/\/mathworld.wolfram.com\/PoincareSeparationTheorem.html), [Random Matrix](https:\/\/mathworld.wolfram.com\/RandomMatrix.html), [Real Matrix](https:\/\/mathworld.wolfram.com\/RealMatrix.html), [Schur's Inequalities](https:\/\/mathworld.wolfram.com\/SchursInequalities.html), [Similarity Transformation](https:\/\/mathworld.wolfram.com\/SimilarityTransformation.html), [Sturmian Separation Theorem](https:\/\/mathworld.wolfram.com\/SturmianSeparationTheorem.html), [Sylvester's Inertia Law](https:\/\/mathworld.wolfram.com\/SylvestersInertiaLaw.html), [Wielandt's Theorem](https:\/\/mathworld.wolfram.com\/WielandtsTheorem.html) [Explore this topic in the MathWorld classroom](https:\/\/mathworld.wolfram.com\/classroom\/Eigenvalue.html)\n\n## Explore with Wolfram\\|Alpha\n## References\nArfken, G. \"Eigenvectors, Eigenvalues.\" §4.7 in *[Mathematical Methods for Physicists, 3rd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0120598760\/ref=nosim\/ericstreasuretro)* Orlando, FL: Academic Press, pp. 229-237, 1985.Hoffman, K. and Kunze, R. \"Characteristic Values.\" §6.2 in *[Linear Algebra, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0135367972\/ref=nosim\/ericstreasuretro)* Englewood Cliffs, NJ: Prentice-Hall, p. 182, 1971.Kaltofen, E. \"Challenges of Symbolic Computation: My Favorite Open Problems.\" *J. Symb. Comput.* **29**, 891-919, 2000.Marcus, M. and Minc, H. *[Introduction to Linear Algebra.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0486656950\/ref=nosim\/ericstreasuretro)* New York: Dover, p. 145, 1988.Nash, J. C. \"The Algebraic Eigenvalue Problem.\" Ch. 9 in *[Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/085274319X\/ref=nosim\/ericstreasuretro)* Bristol, England: Adam Hilger, pp. 102-118, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. \"Eigensystems.\" Ch. 11 in *[Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/052143064X\/ref=nosim\/ericstreasuretro)* Cambridge, England: Cambridge University Press, pp. 449-489, 1992.\n\n## Referenced on Wolfram\\|Alpha\n[Eigenvalue](https:\/\/www.wolframalpha.com\/input\/?i=eigenvalue \"Eigenvalue\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Eigenvalue.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/Eigenvalue.html>\n\n## Subject classifications","meta_canonical":null}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

15 hours ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://mathworld.wolfram.com/Eigenvalue.html
Last Crawled	2026-04-17 08:39:29 (15 hours ago)
First Indexed	2020-04-27 18:30:48 (5 years ago)
HTTP Status Code	200
Meta Title	Eigenvalue -- from Wolfram MathWorld
Meta Description	Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability...
Meta Canonical	null
Boilerpipe Text	Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation ) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector ; there is no analogous distinction between left and right for eigenvalues). The decomposition of a square matrix into eigenvalues and eigenvectors is known in this work as eigen decomposition , and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of is square is known as the eigen decomposition theorem . The Lanczos algorithm is an algorithm for computing the eigenvalues and eigenvectors for large symmetric sparse matrices . Let be a linear transformation represented by a matrix . If there is a vector such that (1) for some scalar , then is called the eigenvalue of with corresponding (right) eigenvector . Letting be a square matrix (2) with eigenvalue , then the corresponding eigenvectors satisfy (3) which is equivalent to the homogeneous system (4) Equation ( 4 ) can be written compactly as (5) where is the identity matrix . As shown in Cramer's rule , a linear system of equations has nontrivial solutions iff the determinant vanishes, so the solutions of equation ( 5 ) are given by (6) This equation is known as the characteristic equation of , and the left-hand side is known as the characteristic polynomial . For example, for a matrix, the eigenvalues are (7) which arises as the solutions of the characteristic equation (8) If all eigenvalues are different, then plugging these back in gives independent equations for the components of each corresponding eigenvector , and the system is said to be nondegenerate. If the eigenvalues are -fold degenerate , then the system is said to be degenerate and the eigenvectors are not linearly independent. In such cases, the additional constraint that the eigenvectors be orthogonal , (9) where is the Kronecker delta , can be applied to yield additional constraints, thus allowing solution for the eigenvectors . Eigenvalues may be computed in the Wolfram Language using Eigenvalues [ matrix ]. Eigenvectors and eigenvalues can be returned together using the command Eigensystem [ matrix ]. Assume we know the eigenvalue for (10) Adding a constant times the identity matrix to , (11) so the new eigenvalues equal the old plus . Multiplying by a constant (12) so the new eigenvalues are the old multiplied by . Now consider a similarity transformation of . Let be the determinant of , then (13) (14) (15) so the eigenvalues are the same as for . See also Brauer's Theorem , Characteristic Equation , Characteristic Polynomial , Complex Matrix , Condition Number , Eigen Decomposition , Eigen Decomposition Theorem , Eigenfunction , Eigenvector , Frobenius Theorem , Gershgorin Circle Theorem , Lanczos Algorithm , Lyapunov's First Theorem , Lyapunov's Second Theorem , Matrix Diagonalization , Ostrowski's Theorem , Perron's Theorem , Perron-Frobenius Theorem , Poincaré Separation Theorem , Random Matrix , Real Matrix , Schur's Inequalities , Similarity Transformation , Sturmian Separation Theorem , Sylvester's Inertia Law , Wielandt's Theorem Explore this topic in the MathWorld classroom Explore with Wolfram\|Alpha References Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 229-237, 1985. Hoffman, K. and Kunze, R. "Characteristic Values." §6.2 in Linear Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 182, 1971. Kaltofen, E. "Challenges of Symbolic Computation: My Favorite Open Problems." J. Symb. Comput. 29 , 891-919, 2000. Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 145, 1988. Nash, J. C. "The Algebraic Eigenvalue Problem." Ch. 9 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 102-118, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Eigensystems." Ch. 11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 449-489, 1992. Referenced on Wolfram\|Alpha Eigenvalue Cite this as: Weisstein, Eric W. "Eigenvalue." From MathWorld --A Wolfram Resource. https://mathworld.wolfram.com/Eigenvalue.html Subject classifications
Markdown	![](https://mathworld.wolfram.com/images/header/menu-icon.png) TOPICS ![](https://mathworld.wolfram.com/images/sidebar/menu-close.png) [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) - [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) - [Linear Algebra](https://mathworld.wolfram.com/topics/LinearAlgebra.html) - [Matrices](https://mathworld.wolfram.com/topics/Matrices.html) - [Matrix Eigenvalues](https://mathworld.wolfram.com/topics/MatrixEigenvalues.html) - [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) - [Linear Algebra](https://mathworld.wolfram.com/topics/LinearAlgebra.html) - [Matrices](https://mathworld.wolfram.com/topics/Matrices.html) - [Matrix Decomposition](https://mathworld.wolfram.com/topics/MatrixDecomposition.html) # Eigenvalue *** [![DOWNLOAD Mathematica Notebook](https://mathworld.wolfram.com/images/entries/download-notebook-icon.png)Download Wolfram Notebook](https://mathworld.wolfram.com/notebooks/LinearAlgebra/Eigenvalue.nb) Eigenvalues are a special set of scalars associated with a [linear system of equations](https://mathworld.wolfram.com/LinearSystemofEquations.html) (i.e., a [matrix equation](https://mathworld.wolfram.com/MatrixEquation.html)) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to [matrix diagonalization](https://mathworld.wolfram.com/MatrixDiagonalization.html) and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called [eigenvector](https://mathworld.wolfram.com/Eigenvector.html) (or, in general, a corresponding [right eigenvector](https://mathworld.wolfram.com/RightEigenvector.html) and a corresponding [left eigenvector](https://mathworld.wolfram.com/LeftEigenvector.html); there is no analogous distinction between left and right for eigenvalues). The decomposition of a [square matrix](https://mathworld.wolfram.com/SquareMatrix.html) ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline1.svg) into eigenvalues and eigenvectors is known in this work as [eigen decomposition](https://mathworld.wolfram.com/EigenDecomposition.html), and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline2.svg) is [square](https://mathworld.wolfram.com/SquareMatrix.html) is known as the [eigen decomposition theorem](https://mathworld.wolfram.com/EigenDecompositionTheorem.html). The [Lanczos algorithm](https://mathworld.wolfram.com/LanczosAlgorithm.html) is an algorithm for computing the eigenvalues and [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) for large [symmetric](https://mathworld.wolfram.com/SymmetricMatrix.html) [sparse matrices](https://mathworld.wolfram.com/SparseMatrix.html). Let ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline3.svg) be a [linear transformation](https://mathworld.wolfram.com/LinearTransformation.html) represented by a [matrix](https://mathworld.wolfram.com/Matrix.html) ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline4.svg). If there is a [vector](https://mathworld.wolfram.com/Vector.html) ![X in R^n!=0](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline5.svg) such that \| \| \| \|---\|---\| \| ![ AX=lambdaX ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation1.svg) \| (1) \| for some [scalar](https://mathworld.wolfram.com/Scalar.html) ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline6.svg), then ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline7.svg) is called the eigenvalue of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline8.svg) with corresponding (right) [eigenvector](https://mathworld.wolfram.com/Eigenvector.html) ![X](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline9.svg). Letting ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline10.svg) be a ![k×k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline11.svg) [square matrix](https://mathworld.wolfram.com/SquareMatrix.html) \| \| \| \|---\|---\| \| ![ \[a\_(11) a\_(12) ... a\_(1k); a\_(21) a\_(22) ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)\] ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation2.svg) \| (2) \| with eigenvalue ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline12.svg), then the corresponding [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) satisfy \| \| \| \|---\|---\| \| ![ \[a\_(11) a\_(12) ... a\_(1k); a\_(21) a\_(22) ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)\]\[x\_1; x\_2; \\|; x\_k\]=lambda\[x\_1; x\_2; \\|; x\_k\], ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation3.svg) \| (3) \| which is equivalent to the homogeneous system \| \| \| \|---\|---\| \| ![ \[a\_(11)-lambda a\_(12) ... a\_(1k); a\_(21) a\_(22)-lambda ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)-lambda\]\[x\_1; x\_2; \\|; x\_k\]=\[0; 0; \\|; 0\]. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation4.svg) \| (4) \| Equation ([4](https://mathworld.wolfram.com/Eigenvalue.html#eqn4)) can be written compactly as \| \| \| \|---\|---\| \| ![ (A-lambdaI)X=0, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation5.svg) \| (5) \| where ![I](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline13.svg) is the [identity matrix](https://mathworld.wolfram.com/IdentityMatrix.html). As shown in [Cramer's rule](https://mathworld.wolfram.com/CramersRule.html), a [linear system of equations](https://mathworld.wolfram.com/LinearSystemofEquations.html) has nontrivial solutions [iff](https://mathworld.wolfram.com/Iff.html) the [determinant](https://mathworld.wolfram.com/Determinant.html) vanishes, so the solutions of equation ([5](https://mathworld.wolfram.com/Eigenvalue.html#eqn5)) are given by \| \| \| \|---\|---\| \| ![ det(A-lambdaI)=0. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation6.svg) \| (6) \| This equation is known as the [characteristic equation](https://mathworld.wolfram.com/CharacteristicEquation.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline14.svg), and the left-hand side is known as the [characteristic polynomial](https://mathworld.wolfram.com/CharacteristicPolynomial.html). For example, for a ![2×2](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline15.svg) matrix, the eigenvalues are \| \| \| \|---\|---\| \| ![ lambda\_+/-=1/2\[(a\_(11)+a\_(22))+/-sqrt(4a\_(12)a\_(21)+(a\_(11)-a\_(22))^2)\], ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation7.svg) \| (7) \| which arises as the solutions of the [characteristic equation](https://mathworld.wolfram.com/CharacteristicEquation.html) \| \| \| \|---\|---\| \| ![ x^2-x(a\_(11)+a\_(22))+(a\_(11)a\_(22)-a\_(12)a\_(21))=0. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation8.svg) \| (8) \| If all ![k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline16.svg) eigenvalues are different, then plugging these back in gives ![k-1](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline17.svg) independent equations for the ![k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline18.svg) components of each corresponding [eigenvector](https://mathworld.wolfram.com/Eigenvector.html), and the system is said to be nondegenerate. If the eigenvalues are ![n](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline19.svg)\-fold [degenerate](https://mathworld.wolfram.com/Degenerate.html), then the system is said to be degenerate and the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) are not linearly independent. In such cases, the additional constraint that the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) be [orthogonal](https://mathworld.wolfram.com/OrthogonalVectors.html), \| \| \| \|---\|---\| \| ![ X\_i·X\_j=\\|X\_i\\|\\|X\_j\\|delta\_(ij), ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation9.svg) \| (9) \| where ![delta\_(ij)](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline20.svg) is the [Kronecker delta](https://mathworld.wolfram.com/KroneckerDelta.html), can be applied to yield ![n](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline21.svg) additional constraints, thus allowing solution for the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html). Eigenvalues may be computed in the [Wolfram Language](http://www.wolfram.com/language/) using [Eigenvalues](http://reference.wolfram.com/language/ref/Eigenvalues.html)\[matrix\]. Eigenvectors and eigenvalues can be returned together using the command [Eigensystem](http://reference.wolfram.com/language/ref/Eigensystem.html)\[matrix\]. Assume we know the eigenvalue for \| \| \| \|---\|---\| \| ![ AX=lambdaX. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation10.svg) \| (10) \| Adding a constant times the [identity matrix](https://mathworld.wolfram.com/IdentityMatrix.html) to ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline22.svg), \| \| \| \|---\|---\| \| ![ (A+cI)X=(lambda+c)X=lambda^'X, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation11.svg) \| (11) \| so the new eigenvalues equal the old plus ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline23.svg). Multiplying ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline24.svg) by a constant ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline25.svg) \| \| \| \|---\|---\| \| ![ (cA)X=c(lambdaX)=lambda^'X, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation12.svg) \| (12) \| so the new eigenvalues are the old multiplied by ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline26.svg). Now consider a [similarity transformation](https://mathworld.wolfram.com/SimilarityTransformation.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline27.svg). Let ![\\|A\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline28.svg) be the [determinant](https://mathworld.wolfram.com/Determinant.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline29.svg), then \| \| \| \| \| \|---\|---\|---\|---\| \| ![\\|Z^(-1)AZ-lambdaI\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline30.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline31.svg) \| ![\\|Z^(-1)(A-lambdaI)Z\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline32.svg) \| (13) \| \| ![](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline33.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline34.svg) \| ![\\|Z\\|\\|A-lambdaI\\|\\|Z^(-1)\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline35.svg) \| (14) \| \| ![](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline36.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline37.svg) \| ![\\|A-lambdaI\\|,](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline38.svg) \| (15) \| so the eigenvalues are the same as for ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline39.svg). *** ## See also [Brauer's Theorem](https://mathworld.wolfram.com/BrauersTheorem.html), [Characteristic Equation](https://mathworld.wolfram.com/CharacteristicEquation.html), [Characteristic Polynomial](https://mathworld.wolfram.com/CharacteristicPolynomial.html), [Complex Matrix](https://mathworld.wolfram.com/ComplexMatrix.html), [Condition Number](https://mathworld.wolfram.com/ConditionNumber.html), [Eigen Decomposition](https://mathworld.wolfram.com/EigenDecomposition.html), [Eigen Decomposition Theorem](https://mathworld.wolfram.com/EigenDecompositionTheorem.html), [Eigenfunction](https://mathworld.wolfram.com/Eigenfunction.html), [Eigenvector](https://mathworld.wolfram.com/Eigenvector.html), [Frobenius Theorem](https://mathworld.wolfram.com/FrobeniusTheorem.html), [Gershgorin Circle Theorem](https://mathworld.wolfram.com/GershgorinCircleTheorem.html), [Lanczos Algorithm](https://mathworld.wolfram.com/LanczosAlgorithm.html), [Lyapunov's First Theorem](https://mathworld.wolfram.com/LyapunovsFirstTheorem.html), [Lyapunov's Second Theorem](https://mathworld.wolfram.com/LyapunovsSecondTheorem.html), [Matrix Diagonalization](https://mathworld.wolfram.com/MatrixDiagonalization.html), [Ostrowski's Theorem](https://mathworld.wolfram.com/OstrowskisTheorem.html), [Perron's Theorem](https://mathworld.wolfram.com/PerronsTheorem.html), [Perron-Frobenius Theorem](https://mathworld.wolfram.com/Perron-FrobeniusTheorem.html), [Poincaré Separation Theorem](https://mathworld.wolfram.com/PoincareSeparationTheorem.html), [Random Matrix](https://mathworld.wolfram.com/RandomMatrix.html), [Real Matrix](https://mathworld.wolfram.com/RealMatrix.html), [Schur's Inequalities](https://mathworld.wolfram.com/SchursInequalities.html), [Similarity Transformation](https://mathworld.wolfram.com/SimilarityTransformation.html), [Sturmian Separation Theorem](https://mathworld.wolfram.com/SturmianSeparationTheorem.html), [Sylvester's Inertia Law](https://mathworld.wolfram.com/SylvestersInertiaLaw.html), [Wielandt's Theorem](https://mathworld.wolfram.com/WielandtsTheorem.html) [Explore this topic in the MathWorld classroom](https://mathworld.wolfram.com/classroom/Eigenvalue.html) ## Explore with Wolfram\\|Alpha ![WolframAlpha](https://mathworld.wolfram.com/images/wolframalpha/WA-logo.png) More things to try: - [eigenvalue calculator](https://www.wolframalpha.com/input/?i=eigenvalue+calculator) - [eigenvalue](https://www.wolframalpha.com/input/?i=eigenvalue) - [eigenvalue {{0,1},{1,0}}](https://www.wolframalpha.com/input/?i=eigenvalue+{{0%2C1}%2C{1%2C0}}) ## References Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in [Mathematical Methods for Physicists, 3rd ed.](http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro) Orlando, FL: Academic Press, pp. 229-237, 1985. Hoffman, K. and Kunze, R. "Characteristic Values." §6.2 in [Linear Algebra, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0135367972/ref=nosim/ericstreasuretro) Englewood Cliffs, NJ: Prentice-Hall, p. 182, 1971. Kaltofen, E. "Challenges of Symbolic Computation: My Favorite Open Problems." J. Symb. Comput. 29, 891-919, 2000. Marcus, M. and Minc, H. [Introduction to Linear Algebra.](http://www.amazon.com/exec/obidos/ASIN/0486656950/ref=nosim/ericstreasuretro) New York: Dover, p. 145, 1988. Nash, J. C. "The Algebraic Eigenvalue Problem." Ch. 9 in [Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/085274319X/ref=nosim/ericstreasuretro) Bristol, England: Adam Hilger, pp. 102-118, 1990. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Eigensystems." Ch. 11 in [Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/052143064X/ref=nosim/ericstreasuretro) Cambridge, England: Cambridge University Press, pp. 449-489, 1992. ## Referenced on Wolfram\\|Alpha [Eigenvalue](https://www.wolframalpha.com/input/?i=eigenvalue "Eigenvalue") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Eigenvalue." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/Eigenvalue.html> ## Subject classifications - [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) - [Linear Algebra](https://mathworld.wolfram.com/topics/LinearAlgebra.html) - [Matrices](https://mathworld.wolfram.com/topics/Matrices.html) - [Matrix Eigenvalues](https://mathworld.wolfram.com/topics/MatrixEigenvalues.html) - [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) - [Linear Algebra](https://mathworld.wolfram.com/topics/LinearAlgebra.html) - [Matrices](https://mathworld.wolfram.com/topics/Matrices.html) - [Matrix Decomposition](https://mathworld.wolfram.com/topics/MatrixDecomposition.html) - [About MathWorld](https://mathworld.wolfram.com/about/) - [MathWorld Classroom](https://mathworld.wolfram.com/classroom/) - [Contribute](https://mathworld.wolfram.com/contact/) - [MathWorld Book](https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20) - [wolfram.com](https://www.wolfram.com/) - [13,311 Entries](https://mathworld.wolfram.com/whatsnew/) - [Last Updated: Wed Mar 25 2026](https://mathworld.wolfram.com/whatsnew/) - [©1999–2026 Wolfram Research, Inc.](https://www.wolfram.com/) - [Terms of Use](https://www.wolfram.com/legal/terms/mathworld.html) - [![Wolfram](https://mathworld.wolfram.com/images/footer/wolfram-logo.png)](https://www.wolfram.com/) - [wolfram.com](https://www.wolfram.com/) - [Wolfram for Education](https://www.wolfram.com/education/) - Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research [Find out if you already have access to Wolfram tech through your organization](https://www.wolfram.com/siteinfo/) ×
Readable Markdown	[Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) *** Eigenvalues are a special set of scalars associated with a [linear system of equations](https://mathworld.wolfram.com/LinearSystemofEquations.html) (i.e., a [matrix equation](https://mathworld.wolfram.com/MatrixEquation.html)) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144). The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to [matrix diagonalization](https://mathworld.wolfram.com/MatrixDiagonalization.html) and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called [eigenvector](https://mathworld.wolfram.com/Eigenvector.html) (or, in general, a corresponding [right eigenvector](https://mathworld.wolfram.com/RightEigenvector.html) and a corresponding [left eigenvector](https://mathworld.wolfram.com/LeftEigenvector.html); there is no analogous distinction between left and right for eigenvalues). The decomposition of a [square matrix](https://mathworld.wolfram.com/SquareMatrix.html) ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline1.svg) into eigenvalues and eigenvectors is known in this work as [eigen decomposition](https://mathworld.wolfram.com/EigenDecomposition.html), and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline2.svg) is [square](https://mathworld.wolfram.com/SquareMatrix.html) is known as the [eigen decomposition theorem](https://mathworld.wolfram.com/EigenDecompositionTheorem.html). The [Lanczos algorithm](https://mathworld.wolfram.com/LanczosAlgorithm.html) is an algorithm for computing the eigenvalues and [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) for large [symmetric](https://mathworld.wolfram.com/SymmetricMatrix.html) [sparse matrices](https://mathworld.wolfram.com/SparseMatrix.html). Let ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline3.svg) be a [linear transformation](https://mathworld.wolfram.com/LinearTransformation.html) represented by a [matrix](https://mathworld.wolfram.com/Matrix.html) ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline4.svg). If there is a [vector](https://mathworld.wolfram.com/Vector.html) ![X in R^n!=0](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline5.svg) such that \| \| \| \|---\|---\| \| ![ AX=lambdaX ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation1.svg) \| (1) \| for some [scalar](https://mathworld.wolfram.com/Scalar.html) ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline6.svg), then ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline7.svg) is called the eigenvalue of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline8.svg) with corresponding (right) [eigenvector](https://mathworld.wolfram.com/Eigenvector.html) ![X](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline9.svg). Letting ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline10.svg) be a ![k×k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline11.svg) [square matrix](https://mathworld.wolfram.com/SquareMatrix.html) \| \| \| \|---\|---\| \| ![ \[a\_(11) a\_(12) ... a\_(1k); a\_(21) a\_(22) ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)\] ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation2.svg) \| (2) \| with eigenvalue ![lambda](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline12.svg), then the corresponding [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) satisfy \| \| \| \|---\|---\| \| ![ \[a\_(11) a\_(12) ... a\_(1k); a\_(21) a\_(22) ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)\]\[x\_1; x\_2; \\|; x\_k\]=lambda\[x\_1; x\_2; \\|; x\_k\], ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation3.svg) \| (3) \| which is equivalent to the homogeneous system \| \| \| \|---\|---\| \| ![ \[a\_(11)-lambda a\_(12) ... a\_(1k); a\_(21) a\_(22)-lambda ... a\_(2k); \\| \\| ... \\|; a\_(k1) a\_(k2) ... a\_(kk)-lambda\]\[x\_1; x\_2; \\|; x\_k\]=\[0; 0; \\|; 0\]. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation4.svg) \| (4) \| Equation ([4](https://mathworld.wolfram.com/Eigenvalue.html#eqn4)) can be written compactly as \| \| \| \|---\|---\| \| ![ (A-lambdaI)X=0, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation5.svg) \| (5) \| where ![I](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline13.svg) is the [identity matrix](https://mathworld.wolfram.com/IdentityMatrix.html). As shown in [Cramer's rule](https://mathworld.wolfram.com/CramersRule.html), a [linear system of equations](https://mathworld.wolfram.com/LinearSystemofEquations.html) has nontrivial solutions [iff](https://mathworld.wolfram.com/Iff.html) the [determinant](https://mathworld.wolfram.com/Determinant.html) vanishes, so the solutions of equation ([5](https://mathworld.wolfram.com/Eigenvalue.html#eqn5)) are given by \| \| \| \|---\|---\| \| ![ det(A-lambdaI)=0. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation6.svg) \| (6) \| This equation is known as the [characteristic equation](https://mathworld.wolfram.com/CharacteristicEquation.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline14.svg), and the left-hand side is known as the [characteristic polynomial](https://mathworld.wolfram.com/CharacteristicPolynomial.html). For example, for a ![2×2](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline15.svg) matrix, the eigenvalues are \| \| \| \|---\|---\| \| ![ lambda\_+/-=1/2\[(a\_(11)+a\_(22))+/-sqrt(4a\_(12)a\_(21)+(a\_(11)-a\_(22))^2)\], ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation7.svg) \| (7) \| which arises as the solutions of the [characteristic equation](https://mathworld.wolfram.com/CharacteristicEquation.html) \| \| \| \|---\|---\| \| ![ x^2-x(a\_(11)+a\_(22))+(a\_(11)a\_(22)-a\_(12)a\_(21))=0. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation8.svg) \| (8) \| If all ![k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline16.svg) eigenvalues are different, then plugging these back in gives ![k-1](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline17.svg) independent equations for the ![k](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline18.svg) components of each corresponding [eigenvector](https://mathworld.wolfram.com/Eigenvector.html), and the system is said to be nondegenerate. If the eigenvalues are ![n](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline19.svg)\-fold [degenerate](https://mathworld.wolfram.com/Degenerate.html), then the system is said to be degenerate and the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) are not linearly independent. In such cases, the additional constraint that the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html) be [orthogonal](https://mathworld.wolfram.com/OrthogonalVectors.html), \| \| \| \|---\|---\| \| ![ X\_i·X\_j=\\|X\_i\\|\\|X\_j\\|delta\_(ij), ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation9.svg) \| (9) \| where ![delta\_(ij)](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline20.svg) is the [Kronecker delta](https://mathworld.wolfram.com/KroneckerDelta.html), can be applied to yield ![n](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline21.svg) additional constraints, thus allowing solution for the [eigenvectors](https://mathworld.wolfram.com/Eigenvector.html). Eigenvalues may be computed in the [Wolfram Language](http://www.wolfram.com/language/) using [Eigenvalues](http://reference.wolfram.com/language/ref/Eigenvalues.html)\[matrix\]. Eigenvectors and eigenvalues can be returned together using the command [Eigensystem](http://reference.wolfram.com/language/ref/Eigensystem.html)\[matrix\]. Assume we know the eigenvalue for \| \| \| \|---\|---\| \| ![ AX=lambdaX. ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation10.svg) \| (10) \| Adding a constant times the [identity matrix](https://mathworld.wolfram.com/IdentityMatrix.html) to ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline22.svg), \| \| \| \|---\|---\| \| ![ (A+cI)X=(lambda+c)X=lambda^'X, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation11.svg) \| (11) \| so the new eigenvalues equal the old plus ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline23.svg). Multiplying ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline24.svg) by a constant ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline25.svg) \| \| \| \|---\|---\| \| ![ (cA)X=c(lambdaX)=lambda^'X, ](https://mathworld.wolfram.com/images/equations/Eigenvalue/NumberedEquation12.svg) \| (12) \| so the new eigenvalues are the old multiplied by ![c](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline26.svg). Now consider a [similarity transformation](https://mathworld.wolfram.com/SimilarityTransformation.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline27.svg). Let ![\\|A\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline28.svg) be the [determinant](https://mathworld.wolfram.com/Determinant.html) of ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline29.svg), then \| \| \| \| \| \|---\|---\|---\|---\| \| ![\\|Z^(-1)AZ-lambdaI\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline30.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline31.svg) \| ![\\|Z^(-1)(A-lambdaI)Z\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline32.svg) \| (13) \| \| ![](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline33.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline34.svg) \| ![\\|Z\\|\\|A-lambdaI\\|\\|Z^(-1)\\|](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline35.svg) \| (14) \| \| ![](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline36.svg) \| ![\=](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline37.svg) \| ![\\|A-lambdaI\\|,](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline38.svg) \| (15) \| so the eigenvalues are the same as for ![A](https://mathworld.wolfram.com/images/equations/Eigenvalue/Inline39.svg). *** ## See also [Brauer's Theorem](https://mathworld.wolfram.com/BrauersTheorem.html), [Characteristic Equation](https://mathworld.wolfram.com/CharacteristicEquation.html), [Characteristic Polynomial](https://mathworld.wolfram.com/CharacteristicPolynomial.html), [Complex Matrix](https://mathworld.wolfram.com/ComplexMatrix.html), [Condition Number](https://mathworld.wolfram.com/ConditionNumber.html), [Eigen Decomposition](https://mathworld.wolfram.com/EigenDecomposition.html), [Eigen Decomposition Theorem](https://mathworld.wolfram.com/EigenDecompositionTheorem.html), [Eigenfunction](https://mathworld.wolfram.com/Eigenfunction.html), [Eigenvector](https://mathworld.wolfram.com/Eigenvector.html), [Frobenius Theorem](https://mathworld.wolfram.com/FrobeniusTheorem.html), [Gershgorin Circle Theorem](https://mathworld.wolfram.com/GershgorinCircleTheorem.html), [Lanczos Algorithm](https://mathworld.wolfram.com/LanczosAlgorithm.html), [Lyapunov's First Theorem](https://mathworld.wolfram.com/LyapunovsFirstTheorem.html), [Lyapunov's Second Theorem](https://mathworld.wolfram.com/LyapunovsSecondTheorem.html), [Matrix Diagonalization](https://mathworld.wolfram.com/MatrixDiagonalization.html), [Ostrowski's Theorem](https://mathworld.wolfram.com/OstrowskisTheorem.html), [Perron's Theorem](https://mathworld.wolfram.com/PerronsTheorem.html), [Perron-Frobenius Theorem](https://mathworld.wolfram.com/Perron-FrobeniusTheorem.html), [Poincaré Separation Theorem](https://mathworld.wolfram.com/PoincareSeparationTheorem.html), [Random Matrix](https://mathworld.wolfram.com/RandomMatrix.html), [Real Matrix](https://mathworld.wolfram.com/RealMatrix.html), [Schur's Inequalities](https://mathworld.wolfram.com/SchursInequalities.html), [Similarity Transformation](https://mathworld.wolfram.com/SimilarityTransformation.html), [Sturmian Separation Theorem](https://mathworld.wolfram.com/SturmianSeparationTheorem.html), [Sylvester's Inertia Law](https://mathworld.wolfram.com/SylvestersInertiaLaw.html), [Wielandt's Theorem](https://mathworld.wolfram.com/WielandtsTheorem.html) [Explore this topic in the MathWorld classroom](https://mathworld.wolfram.com/classroom/Eigenvalue.html) ## Explore with Wolfram\\|Alpha ## References Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in [Mathematical Methods for Physicists, 3rd ed.](http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro) Orlando, FL: Academic Press, pp. 229-237, 1985.Hoffman, K. and Kunze, R. "Characteristic Values." §6.2 in [Linear Algebra, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0135367972/ref=nosim/ericstreasuretro) Englewood Cliffs, NJ: Prentice-Hall, p. 182, 1971.Kaltofen, E. "Challenges of Symbolic Computation: My Favorite Open Problems." J. Symb. Comput. 29, 891-919, 2000.Marcus, M. and Minc, H. [Introduction to Linear Algebra.](http://www.amazon.com/exec/obidos/ASIN/0486656950/ref=nosim/ericstreasuretro) New York: Dover, p. 145, 1988.Nash, J. C. "The Algebraic Eigenvalue Problem." Ch. 9 in [Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/085274319X/ref=nosim/ericstreasuretro) Bristol, England: Adam Hilger, pp. 102-118, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Eigensystems." Ch. 11 in [Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/052143064X/ref=nosim/ericstreasuretro) Cambridge, England: Cambridge University Press, pp. 449-489, 1992. ## Referenced on Wolfram\\|Alpha [Eigenvalue](https://www.wolframalpha.com/input/?i=eigenvalue "Eigenvalue") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Eigenvalue." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/Eigenvalue.html> ## Subject classifications
Shard	184 (laksa)
Root Hash	3744487911316863784
Unparsed URL	com,wolfram!mathworld,/Eigenvalue.html s443