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  -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(\'http://linear.ups.edu/html/section-EE.html\')), getAhrefsUnparsedNoserviceFromURL(\'http://linear.ups.edu/html/section-EE.html\'))) FORMAT JSONEachRow'

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{"found_url":"http:\/\/linear.ups.edu\/html\/section-EE.html","crawl_time":1774821528,"first_indexed_time":1375983908,"http_code":200,"src_unparsed":"edu,ups!linear,\/html\/section-EE.html h80","src_root_hash":"10365611695833771749","history_drop_reason":null,"meta_title":"","meta_descriptions":[],"attrs_boilerpipe_text":"Section EE Eigenvalues and Eigenvectors In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them.  More theoretical properties will be taken up in the next section. Subsection EEM Eigenvalues and Eigenvectors of a Matrix We start with the principal definition for this chapter. Definition  EEM  Eigenvalues and Eigenvectors of a Matrix Suppose that  A A  is a square matrix of size  n n ,  x ≠ 0 x ≠ 0  is a vector in  C n C n , and  λ λ  is a scalar in  C C .   Then we say  x x  is an  eigenvector  of  A A  with  eigenvalue λ λ  if\n A x = λ x A x = λ x Before going any further, perhaps we should convince you that such things ever happen at all.  Understand the next example, but do not concern yourself with where the pieces come from.  We will have methods soon enough to be able to discover these eigenvectors ourselves. Sage EE Eigenvalues and Eigenvectors Example SEE  hints at a number of intriguing properties, and there are many more.  We will explore the general properties of eigenvalues and eigenvectors in  Section PEE , but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors.  First we need a bit of background material on polynomials and matrices. Subsection PM Polynomials and Matrices A polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition).  We never have occasion to divide when computing the value of a polynomial.  So it is with matrices.  We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication.  We do not normally divide matrices (though sometimes we can multiply by an inverse).  If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix.  So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix.  We will demonstrate with an example. Subsection EEE Existence of Eigenvalues and Eigenvectors Before we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it).  Later, in  Theorem MNEM , we will determine the maximum number of eigenvalues a matrix may have. The determinant ( Definition DM ) will be a powerful tool in  Subsection EE.CEE  when it comes time to compute eigenvalues.  However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant.  Sheldon Axler does just that in his book,  Linear Algebra Done Right .  Here and now, we give Axler's “determinant-free” proof that every matrix has an eigenvalue.  The result is not too startling, but the proof is most enjoyable. Theorem  EMHE  Every Matrix Has an Eigenvalue Suppose  A A  is a square matrix.  Then  A A  has at least one eigenvalue. The proof of  Theorem EMHE  is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical.  We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue. Subsection CEE Computing Eigenvalues and Eigenvectors Fortunately, we need not rely on the procedure of  Theorem EMHE  each time we need an eigenvalue.  It is the determinant, and specifically  Theorem SMZD , that provides the main tool for computing eigenvalues.  Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix,\n A x = λ x ⟺ A x − λ I n x = 0 ⟺ ( A − λ I n ) x = 0 A x = λ x ⟺ A x − λ I n x = 0 ⟺ ( A − λ I n ) x = 0 So, for an eigenvalue  λ λ  and associated eigenvector  x ≠ 0 x ≠ 0 , the vector  x x  will be a nonzero element of the null space of  A − λ I n A − λ I n , while the matrix  A − λ I n A − λ I n  will be singular and therefore have zero determinant.  These ideas are made precise in  Theorem EMRCP  and  Theorem EMNS , but for now this brief discussion should suffice as motivation for the following definition and example. Definition  CP  Characteristic Polynomial Suppose that  A A  is a square matrix of size  n n .  Then the  characteristic polynomial  of  A A  is the polynomial  p A ( x ) p A ( x )  defined by\n p A ( x ) = det ( A − x I n ) p A ( x ) = det ( A − x I n ) The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues. Theorem  EMRCP  Eigenvalues of a Matrix are Roots of Characteristic Polynomials Suppose  A A  is a square matrix.\nThen  λ λ  is an eigenvalue of  A A  if and only if  p A ( λ ) = 0 p A ( λ ) = 0 . Let us now turn our attention to the computation of eigenvectors. Definition  EM  Eigenspace of a Matrix Suppose that  A A  is a square matrix and  λ λ  is an eigenvalue of  A A .  Then the  eigenspace  of  A A  for  λ λ ,  E A ( λ ) E A ( λ ) , is the set of all the eigenvectors of  A A  for  λ λ , together with the inclusion of the zero vector. Example SEE  hinted that the set of eigenvectors for a single eigenvalue might have some closure properties, and with the addition of the one eigenvector that is never an eigenvector,  0 0 , we indeed get a whole subspace. Theorem  EMS  Eigenspace for a Matrix is a Subspace Suppose   A A  is a square matrix of size  n n  and  λ λ  is an eigenvalue of  A A .  Then the eigenspace  E A ( λ ) E A ( λ )  is a subspace of the vector space  C n C n . Theorem EMS  tells us that an eigenspace is a subspace (and hence a vector space in its own right).  Our next theorem tells us how to quickly construct this subspace. Theorem  EMNS  Eigenspace of a Matrix is a Null Space Suppose   A A  is a square matrix of size  n n  and  λ λ  is an eigenvalue of  A A .  Then\n E A ( λ ) = N ( A − λ I n ) E A ( λ ) = N ( A − λ I n ) You might notice the close parallels (and differences) between the proofs of  Theorem EMRCP  and  Theorem EMNS .  Since  Theorem EMNS  describes the set of all the eigenvectors of  A A  as a null space we can use techniques such as  Theorem BNS  to provide concise descriptions of eigenspaces.   Theorem EMNS  also provides a trivial proof for  Theorem EMS . Subsection ECEE Examples of Computing Eigenvalues and Eigenvectors There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix.  These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone.  It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots.  These examples are meant to look similar to a concatenation of  Example CPMS3 ,  Example EMS3  and  Example ESMS3 .  First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples. Definition  AME  Algebraic Multiplicity of an Eigenvalue Suppose that  A A  is a square matrix and  λ λ  is an eigenvalue of  A A .  Then the  algebraic multiplicity  of  λ λ ,  α A ( λ ) α A ( λ ) , is the highest power of  ( x − λ ) ( x − λ )  that divides the characteristic polynomial,  p A ( x ) p A ( x ) . Since an eigenvalue  λ λ  is a root of the characteristic polynomial, there is always a factor of  ( x − λ ) ( x − λ ) , and the algebraic multiplicity is just the power of this factor in a factorization of  p A ( x ) p A ( x ) .  So in particular,  α A ( λ ) ≥ 1 α A ( λ ) ≥ 1 .  Compare the definition of algebraic multiplicity with the next definition. Definition  GME  Geometric Multiplicity of an Eigenvalue Suppose that  A A  is a square matrix and  λ λ  is an eigenvalue of  A A .  Then the  geometric multiplicity  of  λ λ ,  γ A ( λ ) γ A ( λ ) , is the dimension of the eigenspace  E A ( λ ) E A ( λ ) . Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so  γ A ( λ ) ≥ 1 γ A ( λ ) ≥ 1 .","attrs_markdown":"[![](http:\/\/linear.ups.edu\/html\/images\/cover-84x120.png)](http:\/\/linear.pugetsound.edu\/)\n\n[A First Course in Linear Algebra](http:\/\/linear.ups.edu\/html\/fcla.html) » [Eigenvalues](http:\/\/linear.ups.edu\/html\/chapter-E.html) » [Eigenvalues and Eigenvectors](http:\/\/linear.ups.edu\/html\/section-EE.html) »\n\nEigenvalues and Eigenvectors\n\n## [A First Course in Linear Algebra](http:\/\/linear.ups.edu\/html\/fcla.html)\n- [Preface](http:\/\/linear.ups.edu\/html\/preface.html)\n- [Dedication and Acknowledgements](http:\/\/linear.ups.edu\/html\/acknowledgements.html)\n## [Systems of Linear Equations](http:\/\/linear.ups.edu\/html\/chapter-SLE.html)\n- [What is Linear Algebra?](http:\/\/linear.ups.edu\/html\/section-WILA.html)\n- [Solving Systems of Linear Equations](http:\/\/linear.ups.edu\/html\/section-SSLE.html)\n- [Reduced Row-Echelon Form](http:\/\/linear.ups.edu\/html\/section-RREF.html)\n- [Types of Solution Sets](http:\/\/linear.ups.edu\/html\/section-TSS.html)\n- [Homogeneous Systems of Equations](http:\/\/linear.ups.edu\/html\/section-HSE.html)\n- [Nonsingular Matrices](http:\/\/linear.ups.edu\/html\/section-NM.html)\n## [Vectors](http:\/\/linear.ups.edu\/html\/chapter-V.html)\n- [Vector Operations](http:\/\/linear.ups.edu\/html\/section-VO.html)\n- [Linear Combinations](http:\/\/linear.ups.edu\/html\/section-LC.html)\n- [Spanning Sets](http:\/\/linear.ups.edu\/html\/section-SS.html)\n- [Linear Independence](http:\/\/linear.ups.edu\/html\/section-LI.html)\n- [Linear Dependence and Spans](http:\/\/linear.ups.edu\/html\/section-LDS.html)\n- [Orthogonality](http:\/\/linear.ups.edu\/html\/section-O.html)\n## [Matrices](http:\/\/linear.ups.edu\/html\/chapter-M.html)\n- [Matrix Operations](http:\/\/linear.ups.edu\/html\/section-MO.html)\n- [Matrix Multiplication](http:\/\/linear.ups.edu\/html\/section-MM.html)\n- [Matrix Inverses and Systems of Linear Equations](http:\/\/linear.ups.edu\/html\/section-MISLE.html)\n- [Matrix Inverses and Nonsingular Matrices](http:\/\/linear.ups.edu\/html\/section-MINM.html)\n- [Column and Row Spaces](http:\/\/linear.ups.edu\/html\/section-CRS.html)\n- [Four Subsets](http:\/\/linear.ups.edu\/html\/section-FS.html)\n## [Vector Spaces](http:\/\/linear.ups.edu\/html\/chapter-VS.html)\n- [Vector Spaces](http:\/\/linear.ups.edu\/html\/section-VS.html)\n- [Subspaces](http:\/\/linear.ups.edu\/html\/section-S.html)\n- [Linear Independence and Spanning Sets](http:\/\/linear.ups.edu\/html\/section-LISS.html)\n- [Bases](http:\/\/linear.ups.edu\/html\/section-B.html)\n- [Dimension](http:\/\/linear.ups.edu\/html\/section-D.html)\n- [Properties of Dimension](http:\/\/linear.ups.edu\/html\/section-PD.html)\n## [Determinants](http:\/\/linear.ups.edu\/html\/chapter-D.html)\n- [Determinant of a Matrix](http:\/\/linear.ups.edu\/html\/section-DM.html)\n- [Properties of Determinants of Matrices](http:\/\/linear.ups.edu\/html\/section-PDM.html)\n## [Eigenvalues](http:\/\/linear.ups.edu\/html\/chapter-E.html)\n- [Eigenvalues and Eigenvectors](http:\/\/linear.ups.edu\/html\/section-EE.html)\n- [Properties of Eigenvalues and Eigenvectors](http:\/\/linear.ups.edu\/html\/section-PEE.html)\n- [Similarity and Diagonalization](http:\/\/linear.ups.edu\/html\/section-SD.html)\n## [Linear Transformations](http:\/\/linear.ups.edu\/html\/chapter-LT.html)\n- [Linear Transformations](http:\/\/linear.ups.edu\/html\/section-LT.html)\n- [Injective Linear Transformations](http:\/\/linear.ups.edu\/html\/section-ILT.html)\n- [Surjective Linear Transformations](http:\/\/linear.ups.edu\/html\/section-SLT.html)\n- [Invertible Linear Transformations](http:\/\/linear.ups.edu\/html\/section-IVLT.html)\n## [Representations](http:\/\/linear.ups.edu\/html\/chapter-R.html)\n- [Vector Representations](http:\/\/linear.ups.edu\/html\/section-VR.html)\n- [Matrix Representations](http:\/\/linear.ups.edu\/html\/section-MR.html)\n- [Change of Basis](http:\/\/linear.ups.edu\/html\/section-CB.html)\n- [Orthonormal Diagonalization](http:\/\/linear.ups.edu\/html\/section-OD.html)\n## [Preliminaries](http:\/\/linear.ups.edu\/html\/chapter-P.html)\n- [Complex Number Operations](http:\/\/linear.ups.edu\/html\/section-CNO.html)\n- [Sets](http:\/\/linear.ups.edu\/html\/section-SET.html)\n## [Archetypes](http:\/\/linear.ups.edu\/html\/archetypes.html)\n- [A](http:\/\/linear.ups.edu\/html\/archetype-A.html)[B](http:\/\/linear.ups.edu\/html\/archetype-B.html)[C](http:\/\/linear.ups.edu\/html\/archetype-C.html)[D](http:\/\/linear.ups.edu\/html\/archetype-D.html)[E](http:\/\/linear.ups.edu\/html\/archetype-E.html)[F](http:\/\/linear.ups.edu\/html\/archetype-F.html)[G](http:\/\/linear.ups.edu\/html\/archetype-G.html)[H](http:\/\/linear.ups.edu\/html\/archetype-H.html)[I](http:\/\/linear.ups.edu\/html\/archetype-I.html)[J](http:\/\/linear.ups.edu\/html\/archetype-J.html)[K](http:\/\/linear.ups.edu\/html\/archetype-K.html)[L](http:\/\/linear.ups.edu\/html\/archetype-L.html)[M](http:\/\/linear.ups.edu\/html\/archetype-M.html)\n- [N](http:\/\/linear.ups.edu\/html\/archetype-N.html)[O](http:\/\/linear.ups.edu\/html\/archetype-O.html)[P](http:\/\/linear.ups.edu\/html\/archetype-P.html)[Q](http:\/\/linear.ups.edu\/html\/archetype-Q.html)[R](http:\/\/linear.ups.edu\/html\/archetype-R.html)[S](http:\/\/linear.ups.edu\/html\/archetype-S.html)[T](http:\/\/linear.ups.edu\/html\/archetype-T.html)[U](http:\/\/linear.ups.edu\/html\/archetype-U.html)[V](http:\/\/linear.ups.edu\/html\/archetype-V.html)[W](http:\/\/linear.ups.edu\/html\/archetype-W.html)[X](http:\/\/linear.ups.edu\/html\/archetype-X.html)\n## [Reference](http:\/\/linear.ups.edu\/html\/reference.html)\n- [Notation](http:\/\/linear.ups.edu\/html\/notation.html)\n- [Definitions](http:\/\/linear.ups.edu\/html\/definitions.html)\n- [Theorems](http:\/\/linear.ups.edu\/html\/theorems.html)\n- [Diagrams](http:\/\/linear.ups.edu\/html\/diagrams.html)\n- [Examples](http:\/\/linear.ups.edu\/html\/examples.html)\n- [Sage](http:\/\/linear.ups.edu\/html\/sage.html)\n- [Proof Techniques](http:\/\/linear.ups.edu\/html\/techniques.html)\n- [GFDL License](http:\/\/linear.ups.edu\/html\/GFDL.html)\n\n### Section EE Eigenvalues and Eigenvectors\nIn this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.\n\n#### Subsection EEM Eigenvalues and Eigenvectors of a Matrix\nWe start with the principal definition for this chapter.\n\n##### Definition EEM  Eigenvalues and Eigenvectors of a Matrix\nSuppose that A A is a square matrix of size n n, x≠0 x ≠ 0 is a vector in Cn C n, and λ λ is a scalar in C C. Then we say x x is an *eigenvector* of A A with *eigenvalue* λ λ if Ax\\=λx A x \\= λ x\n\nBefore going any further, perhaps we should convince you that such things ever happen at all. Understand the next example, but do not concern yourself with where the pieces come from. We will have methods soon enough to be able to discover these eigenvectors ourselves.\n\n##### [Example SEE](http:\/\/linear.ups.edu\/html\/section-EE.html) Some eigenvalues and eigenvectors\n##### [Sage EE](http:\/\/linear.ups.edu\/html\/section-EE.html) Eigenvalues and Eigenvectors\n[Example SEE](http:\/\/linear.ups.edu\/html\/section-EE.html \"Some eigenvalues and eigenvectors\") hints at a number of intriguing properties, and there are many more. We will explore the general properties of eigenvalues and eigenvectors in [Section PEE](http:\/\/linear.ups.edu\/html\/section-PEE.html \"Properties of Eigenvalues and Eigenvectors\"), but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors. First we need a bit of background material on polynomials and matrices.\n\n#### Subsection PM Polynomials and Matrices\nA polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition). We never have occasion to divide when computing the value of a polynomial. So it is with matrices. We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication. We do not normally divide matrices (though sometimes we can multiply by an inverse). If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix. So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix. We will demonstrate with an example.\n\n##### [Example PM](http:\/\/linear.ups.edu\/html\/section-EE.html) Polynomial of a matrix\n\n#### Subsection EEE Existence of Eigenvalues and Eigenvectors\nBefore we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it). Later, in [Theorem MNEM](http:\/\/linear.ups.edu\/html\/section-EE.html \"Maximum Number of Eigenvalues of a Matrix\"), we will determine the maximum number of eigenvalues a matrix may have.\n\nThe determinant ([Definition DM](http:\/\/linear.ups.edu\/html\/section-EE.html \"Determinant of a Matrix\")) will be a powerful tool in [Subsection EE.CEE](http:\/\/linear.ups.edu\/html\/section-EE.html) when it comes time to compute eigenvalues. However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant. Sheldon Axler does just that in his book, *Linear Algebra Done Right*. Here and now, we give Axler's “determinant-free” proof that every matrix has an eigenvalue. The result is not too startling, but the proof is most enjoyable.\n\n##### Theorem EMHE Every Matrix Has an Eigenvalue\nSuppose A A is a square matrix. Then A A has at least one eigenvalue.\n\n[Proof](http:\/\/linear.ups.edu\/html\/section-EE.html)\n\nThe proof of [Theorem EMHE](http:\/\/linear.ups.edu\/html\/section-EE.html \"Every Matrix Has an Eigenvalue\") is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical. We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue.\n\n##### [Example CAEHW](http:\/\/linear.ups.edu\/html\/section-EE.html) Computing an eigenvalue the hard way\n\n#### Subsection CEE Computing Eigenvalues and Eigenvectors\nFortunately, we need not rely on the procedure of [Theorem EMHE](http:\/\/linear.ups.edu\/html\/section-EE.html \"Every Matrix Has an Eigenvalue\") each time we need an eigenvalue. It is the determinant, and specifically [Theorem SMZD](http:\/\/linear.ups.edu\/html\/section-EE.html \"Singular Matrices have Zero Determinants\"), that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, Ax\\=λx⟺Ax−λInx\\=0⟺(A−λIn)x\\=0 A x \\= λ x ⟺ A x − λ I n x \\= 0 ⟺ ( A − λ I n ) x \\= 0\n\nSo, for an eigenvalue λ λ and associated eigenvector x≠0 x ≠ 0, the vector x x will be a nonzero element of the null space of A−λIn A − λ I n, while the matrix A−λIn A − λ I n will be singular and therefore have zero determinant. These ideas are made precise in [Theorem EMRCP](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenvalues of a Matrix are Roots of Characteristic Polynomials\") and [Theorem EMNS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace of a Matrix is a Null Space\"), but for now this brief discussion should suffice as motivation for the following definition and example.\n\n##### Definition CP  Characteristic Polynomial\nSuppose that A A is a square matrix of size n n. Then the *characteristic polynomial* of A A is the polynomial pA(x) p A ( x ) defined by pA(x)\\=det(A−xIn) p A ( x ) \\= det ( A − x I n )\n##### [Example CPMS3](http:\/\/linear.ups.edu\/html\/section-EE.html) Characteristic polynomial of a matrix, size 3\nThe characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues.\n\n##### Theorem EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials\nSuppose A A is a square matrix. Then λ λ is an eigenvalue of A A if and only if pA(λ)\\=0 p A ( λ ) \\= 0.\n\n[Proof](http:\/\/linear.ups.edu\/html\/section-EE.html)\n##### [Example EMS3](http:\/\/linear.ups.edu\/html\/section-EE.html) Eigenvalues of a matrix, size 3\nLet us now turn our attention to the computation of eigenvectors.\n\n##### Definition EM  Eigenspace of a Matrix\nSuppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the *eigenspace* of A A for λ λ, EA(λ) E A ( λ ), is the set of all the eigenvectors of A A for λ λ, together with the inclusion of the zero vector.\n\n[Example SEE](http:\/\/linear.ups.edu\/html\/section-EE.html \"Some eigenvalues and eigenvectors\") hinted that the set of eigenvectors for a single eigenvalue might have some closure properties, and with the addition of the one eigenvector that is never an eigenvector, 0 0, we indeed get a whole subspace.\n\n##### Theorem EMS Eigenspace for a Matrix is a Subspace\nSuppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A. Then the eigenspace EA(λ) E A ( λ ) is a subspace of the vector space Cn C n.\n\n[Proof](http:\/\/linear.ups.edu\/html\/section-EE.html)\n\n[Theorem EMS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace for a Matrix is a Subspace\") tells us that an eigenspace is a subspace (and hence a vector space in its own right). Our next theorem tells us how to quickly construct this subspace.\n\n##### Theorem EMNS Eigenspace of a Matrix is a Null Space\nSuppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A. Then EA(λ)\\=N(A−λIn) E A ( λ ) \\= N ( A − λ I n )\n\n[Proof](http:\/\/linear.ups.edu\/html\/section-EE.html)\n\nYou might notice the close parallels (and differences) between the proofs of [Theorem EMRCP](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenvalues of a Matrix are Roots of Characteristic Polynomials\") and [Theorem EMNS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace of a Matrix is a Null Space\"). Since [Theorem EMNS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace of a Matrix is a Null Space\") describes the set of all the eigenvectors of A A as a null space we can use techniques such as [Theorem BNS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Basis for Null Spaces\") to provide concise descriptions of eigenspaces. [Theorem EMNS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace of a Matrix is a Null Space\") also provides a trivial proof for [Theorem EMS](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspace for a Matrix is a Subspace\").\n\n##### [Example ESMS3](http:\/\/linear.ups.edu\/html\/section-EE.html) Eigenspaces of a matrix, size 3\n\n#### Subsection ECEE Examples of Computing Eigenvalues and Eigenvectors\nThere are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of [Example CPMS3](http:\/\/linear.ups.edu\/html\/section-EE.html \"Characteristic polynomial of a matrix, size 3\"), [Example EMS3](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenvalues of a matrix, size 3\") and [Example ESMS3](http:\/\/linear.ups.edu\/html\/section-EE.html \"Eigenspaces of a matrix, size 3\"). First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples.\n\n##### Definition AME  Algebraic Multiplicity of an Eigenvalue\nSuppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the *algebraic multiplicity* of λ λ, αA(λ) α A ( λ ), is the highest power of (x−λ) ( x − λ ) that divides the characteristic polynomial, pA(x) p A ( x ).\n\nSince an eigenvalue λ λ is a root of the characteristic polynomial, there is always a factor of (x−λ) ( x − λ ), and the algebraic multiplicity is just the power of this factor in a factorization of pA(x) p A ( x ). So in particular, αA(λ)≥1 α A ( λ ) ≥ 1. Compare the definition of algebraic multiplicity with the next definition.\n\n##### Definition GME  Geometric Multiplicity of an Eigenvalue\nSuppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the *geometric multiplicity* of λ λ, γA(λ) γ A ( λ ), is the dimension of the eigenspace EA(λ) E A ( λ ).\n\nEvery eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so γA(λ)≥1 γ A ( λ ) ≥ 1.\n\n##### [Example EMMS4](http:\/\/linear.ups.edu\/html\/section-EE.html) Eigenvalue multiplicities, matrix of size 4\n##### [Example ESMS4](http:\/\/linear.ups.edu\/html\/section-EE.html) Eigenvalues, symmetric matrix of size 4\n##### [Example HMEM5](http:\/\/linear.ups.edu\/html\/section-EE.html) High multiplicity eigenvalues, matrix of size 5\n##### [Example CEMS6](http:\/\/linear.ups.edu\/html\/section-EE.html) Complex eigenvalues, matrix of size 6\n##### [Example DEMS5](http:\/\/linear.ups.edu\/html\/section-EE.html) Distinct eigenvalues, matrix of size 5\n##### [Sage CEVAL](http:\/\/linear.ups.edu\/html\/section-EE.html) Computing Eigenvalues\n##### [Sage CEVEC](http:\/\/linear.ups.edu\/html\/section-EE.html) Computing Eigenvectors\n#### [Reading Questions](http:\/\/linear.ups.edu\/html\/section-EE.html)\n#### [Exercises](http:\/\/linear.ups.edu\/html\/section-EE.html)\n\n[![web analytics](http:\/\/c.statcounter.com\/8375157\/0\/c03f6ece\/1\/)](http:\/\/statcounter.com\/ \"web analytics\")","attrs_readable_markdown":null,"meta_canonical":null}

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Boilerpipe Text	Section EE Eigenvalues and Eigenvectors In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section. Subsection EEM Eigenvalues and Eigenvectors of a Matrix We start with the principal definition for this chapter. Definition EEM Eigenvalues and Eigenvectors of a Matrix Suppose that A A is a square matrix of size n n , x ≠ 0 x ≠ 0 is a vector in C n C n , and λ λ is a scalar in C C . Then we say x x is an eigenvector of A A with eigenvalue λ λ if A x = λ x A x = λ x Before going any further, perhaps we should convince you that such things ever happen at all. Understand the next example, but do not concern yourself with where the pieces come from. We will have methods soon enough to be able to discover these eigenvectors ourselves. Sage EE Eigenvalues and Eigenvectors Example SEE hints at a number of intriguing properties, and there are many more. We will explore the general properties of eigenvalues and eigenvectors in Section PEE , but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors. First we need a bit of background material on polynomials and matrices. Subsection PM Polynomials and Matrices A polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition). We never have occasion to divide when computing the value of a polynomial. So it is with matrices. We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication. We do not normally divide matrices (though sometimes we can multiply by an inverse). If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix. So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix. We will demonstrate with an example. Subsection EEE Existence of Eigenvalues and Eigenvectors Before we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it). Later, in Theorem MNEM , we will determine the maximum number of eigenvalues a matrix may have. The determinant ( Definition DM ) will be a powerful tool in Subsection EE.CEE when it comes time to compute eigenvalues. However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant. Sheldon Axler does just that in his book, Linear Algebra Done Right . Here and now, we give Axler's “determinant-free” proof that every matrix has an eigenvalue. The result is not too startling, but the proof is most enjoyable. Theorem EMHE Every Matrix Has an Eigenvalue Suppose A A is a square matrix. Then A A has at least one eigenvalue. The proof of Theorem EMHE is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical. We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue. Subsection CEE Computing Eigenvalues and Eigenvectors Fortunately, we need not rely on the procedure of Theorem EMHE each time we need an eigenvalue. It is the determinant, and specifically Theorem SMZD , that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, A x = λ x ⟺ A x − λ I n x = 0 ⟺ ( A − λ I n ) x = 0 A x = λ x ⟺ A x − λ I n x = 0 ⟺ ( A − λ I n ) x = 0 So, for an eigenvalue λ λ and associated eigenvector x ≠ 0 x ≠ 0 , the vector x x will be a nonzero element of the null space of A − λ I n A − λ I n , while the matrix A − λ I n A − λ I n will be singular and therefore have zero determinant. These ideas are made precise in Theorem EMRCP and Theorem EMNS , but for now this brief discussion should suffice as motivation for the following definition and example. Definition CP Characteristic Polynomial Suppose that A A is a square matrix of size n n . Then the characteristic polynomial of A A is the polynomial p A ( x ) p A ( x ) defined by p A ( x ) = det ( A − x I n ) p A ( x ) = det ( A − x I n ) The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues. Theorem EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials Suppose A A is a square matrix. Then λ λ is an eigenvalue of A A if and only if p A ( λ ) = 0 p A ( λ ) = 0 . Let us now turn our attention to the computation of eigenvectors. Definition EM Eigenspace of a Matrix Suppose that A A is a square matrix and λ λ is an eigenvalue of A A . Then the eigenspace of A A for λ λ , E A ( λ ) E A ( λ ) , is the set of all the eigenvectors of A A for λ λ , together with the inclusion of the zero vector. Example SEE hinted that the set of eigenvectors for a single eigenvalue might have some closure properties, and with the addition of the one eigenvector that is never an eigenvector, 0 0 , we indeed get a whole subspace. Theorem EMS Eigenspace for a Matrix is a Subspace Suppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A . Then the eigenspace E A ( λ ) E A ( λ ) is a subspace of the vector space C n C n . Theorem EMS tells us that an eigenspace is a subspace (and hence a vector space in its own right). Our next theorem tells us how to quickly construct this subspace. Theorem EMNS Eigenspace of a Matrix is a Null Space Suppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A . Then E A ( λ ) = N ( A − λ I n ) E A ( λ ) = N ( A − λ I n ) You might notice the close parallels (and differences) between the proofs of Theorem EMRCP and Theorem EMNS . Since Theorem EMNS describes the set of all the eigenvectors of A A as a null space we can use techniques such as Theorem BNS to provide concise descriptions of eigenspaces. Theorem EMNS also provides a trivial proof for Theorem EMS . Subsection ECEE Examples of Computing Eigenvalues and Eigenvectors There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of Example CPMS3 , Example EMS3 and Example ESMS3 . First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples. Definition AME Algebraic Multiplicity of an Eigenvalue Suppose that A A is a square matrix and λ λ is an eigenvalue of A A . Then the algebraic multiplicity of λ λ , α A ( λ ) α A ( λ ) , is the highest power of ( x − λ ) ( x − λ ) that divides the characteristic polynomial, p A ( x ) p A ( x ) . Since an eigenvalue λ λ is a root of the characteristic polynomial, there is always a factor of ( x − λ ) ( x − λ ) , and the algebraic multiplicity is just the power of this factor in a factorization of p A ( x ) p A ( x ) . So in particular, α A ( λ ) ≥ 1 α A ( λ ) ≥ 1 . Compare the definition of algebraic multiplicity with the next definition. Definition GME Geometric Multiplicity of an Eigenvalue Suppose that A A is a square matrix and λ λ is an eigenvalue of A A . Then the geometric multiplicity of λ λ , γ A ( λ ) γ A ( λ ) , is the dimension of the eigenspace E A ( λ ) E A ( λ ) . Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so γ A ( λ ) ≥ 1 γ A ( λ ) ≥ 1 .
Markdown	[![](http://linear.ups.edu/html/images/cover-84x120.png)](http://linear.pugetsound.edu/) [A First Course in Linear Algebra](http://linear.ups.edu/html/fcla.html) » [Eigenvalues](http://linear.ups.edu/html/chapter-E.html) » [Eigenvalues and Eigenvectors](http://linear.ups.edu/html/section-EE.html) » Eigenvalues and Eigenvectors ## [A First Course in Linear Algebra](http://linear.ups.edu/html/fcla.html) - [Preface](http://linear.ups.edu/html/preface.html) - [Dedication and Acknowledgements](http://linear.ups.edu/html/acknowledgements.html) ## [Systems of Linear Equations](http://linear.ups.edu/html/chapter-SLE.html) - [What is Linear Algebra?](http://linear.ups.edu/html/section-WILA.html) - [Solving Systems of Linear Equations](http://linear.ups.edu/html/section-SSLE.html) - [Reduced Row-Echelon Form](http://linear.ups.edu/html/section-RREF.html) - [Types of Solution Sets](http://linear.ups.edu/html/section-TSS.html) - [Homogeneous Systems of Equations](http://linear.ups.edu/html/section-HSE.html) - [Nonsingular Matrices](http://linear.ups.edu/html/section-NM.html) ## [Vectors](http://linear.ups.edu/html/chapter-V.html) - [Vector Operations](http://linear.ups.edu/html/section-VO.html) - [Linear Combinations](http://linear.ups.edu/html/section-LC.html) - [Spanning Sets](http://linear.ups.edu/html/section-SS.html) - [Linear Independence](http://linear.ups.edu/html/section-LI.html) - [Linear Dependence and Spans](http://linear.ups.edu/html/section-LDS.html) - [Orthogonality](http://linear.ups.edu/html/section-O.html) ## [Matrices](http://linear.ups.edu/html/chapter-M.html) - [Matrix Operations](http://linear.ups.edu/html/section-MO.html) - [Matrix Multiplication](http://linear.ups.edu/html/section-MM.html) - [Matrix Inverses and Systems of Linear Equations](http://linear.ups.edu/html/section-MISLE.html) - [Matrix Inverses and Nonsingular Matrices](http://linear.ups.edu/html/section-MINM.html) - [Column and Row Spaces](http://linear.ups.edu/html/section-CRS.html) - [Four Subsets](http://linear.ups.edu/html/section-FS.html) ## [Vector Spaces](http://linear.ups.edu/html/chapter-VS.html) - [Vector Spaces](http://linear.ups.edu/html/section-VS.html) - [Subspaces](http://linear.ups.edu/html/section-S.html) - [Linear Independence and Spanning Sets](http://linear.ups.edu/html/section-LISS.html) - [Bases](http://linear.ups.edu/html/section-B.html) - [Dimension](http://linear.ups.edu/html/section-D.html) - [Properties of Dimension](http://linear.ups.edu/html/section-PD.html) ## [Determinants](http://linear.ups.edu/html/chapter-D.html) - [Determinant of a Matrix](http://linear.ups.edu/html/section-DM.html) - [Properties of Determinants of Matrices](http://linear.ups.edu/html/section-PDM.html) ## [Eigenvalues](http://linear.ups.edu/html/chapter-E.html) - [Eigenvalues and Eigenvectors](http://linear.ups.edu/html/section-EE.html) - [Properties of Eigenvalues and Eigenvectors](http://linear.ups.edu/html/section-PEE.html) - [Similarity and Diagonalization](http://linear.ups.edu/html/section-SD.html) ## [Linear Transformations](http://linear.ups.edu/html/chapter-LT.html) - [Linear Transformations](http://linear.ups.edu/html/section-LT.html) - [Injective Linear Transformations](http://linear.ups.edu/html/section-ILT.html) - [Surjective Linear Transformations](http://linear.ups.edu/html/section-SLT.html) - [Invertible Linear Transformations](http://linear.ups.edu/html/section-IVLT.html) ## [Representations](http://linear.ups.edu/html/chapter-R.html) - [Vector Representations](http://linear.ups.edu/html/section-VR.html) - [Matrix Representations](http://linear.ups.edu/html/section-MR.html) - [Change of Basis](http://linear.ups.edu/html/section-CB.html) - [Orthonormal Diagonalization](http://linear.ups.edu/html/section-OD.html) ## [Preliminaries](http://linear.ups.edu/html/chapter-P.html) - [Complex Number Operations](http://linear.ups.edu/html/section-CNO.html) - [Sets](http://linear.ups.edu/html/section-SET.html) ## [Archetypes](http://linear.ups.edu/html/archetypes.html) - [A](http://linear.ups.edu/html/archetype-A.html)[B](http://linear.ups.edu/html/archetype-B.html)[C](http://linear.ups.edu/html/archetype-C.html)[D](http://linear.ups.edu/html/archetype-D.html)[E](http://linear.ups.edu/html/archetype-E.html)[F](http://linear.ups.edu/html/archetype-F.html)[G](http://linear.ups.edu/html/archetype-G.html)[H](http://linear.ups.edu/html/archetype-H.html)[I](http://linear.ups.edu/html/archetype-I.html)[J](http://linear.ups.edu/html/archetype-J.html)[K](http://linear.ups.edu/html/archetype-K.html)[L](http://linear.ups.edu/html/archetype-L.html)[M](http://linear.ups.edu/html/archetype-M.html) - [N](http://linear.ups.edu/html/archetype-N.html)[O](http://linear.ups.edu/html/archetype-O.html)[P](http://linear.ups.edu/html/archetype-P.html)[Q](http://linear.ups.edu/html/archetype-Q.html)[R](http://linear.ups.edu/html/archetype-R.html)[S](http://linear.ups.edu/html/archetype-S.html)[T](http://linear.ups.edu/html/archetype-T.html)[U](http://linear.ups.edu/html/archetype-U.html)[V](http://linear.ups.edu/html/archetype-V.html)[W](http://linear.ups.edu/html/archetype-W.html)[X](http://linear.ups.edu/html/archetype-X.html) ## [Reference](http://linear.ups.edu/html/reference.html) - [Notation](http://linear.ups.edu/html/notation.html) - [Definitions](http://linear.ups.edu/html/definitions.html) - [Theorems](http://linear.ups.edu/html/theorems.html) - [Diagrams](http://linear.ups.edu/html/diagrams.html) - [Examples](http://linear.ups.edu/html/examples.html) - [Sage](http://linear.ups.edu/html/sage.html) - [Proof Techniques](http://linear.ups.edu/html/techniques.html) - [GFDL License](http://linear.ups.edu/html/GFDL.html) ### Section EE Eigenvalues and Eigenvectors In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section. #### Subsection EEM Eigenvalues and Eigenvectors of a Matrix We start with the principal definition for this chapter. ##### Definition EEM Eigenvalues and Eigenvectors of a Matrix Suppose that A A is a square matrix of size n n, x≠0 x ≠ 0 is a vector in Cn C n, and λ λ is a scalar in C C. Then we say x x is an eigenvector of A A with eigenvalue λ λ if Ax\=λx A x \= λ x Before going any further, perhaps we should convince you that such things ever happen at all. Understand the next example, but do not concern yourself with where the pieces come from. We will have methods soon enough to be able to discover these eigenvectors ourselves. ##### [Example SEE](http://linear.ups.edu/html/section-EE.html) Some eigenvalues and eigenvectors ##### [Sage EE](http://linear.ups.edu/html/section-EE.html) Eigenvalues and Eigenvectors [Example SEE](http://linear.ups.edu/html/section-EE.html "Some eigenvalues and eigenvectors") hints at a number of intriguing properties, and there are many more. We will explore the general properties of eigenvalues and eigenvectors in [Section PEE](http://linear.ups.edu/html/section-PEE.html "Properties of Eigenvalues and Eigenvectors"), but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors. First we need a bit of background material on polynomials and matrices. #### Subsection PM Polynomials and Matrices A polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition). We never have occasion to divide when computing the value of a polynomial. So it is with matrices. We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication. We do not normally divide matrices (though sometimes we can multiply by an inverse). If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix. So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix. We will demonstrate with an example. ##### [Example PM](http://linear.ups.edu/html/section-EE.html) Polynomial of a matrix #### Subsection EEE Existence of Eigenvalues and Eigenvectors Before we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it). Later, in [Theorem MNEM](http://linear.ups.edu/html/section-EE.html "Maximum Number of Eigenvalues of a Matrix"), we will determine the maximum number of eigenvalues a matrix may have. The determinant ([Definition DM](http://linear.ups.edu/html/section-EE.html "Determinant of a Matrix")) will be a powerful tool in [Subsection EE.CEE](http://linear.ups.edu/html/section-EE.html) when it comes time to compute eigenvalues. However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant. Sheldon Axler does just that in his book, Linear Algebra Done Right. Here and now, we give Axler's “determinant-free” proof that every matrix has an eigenvalue. The result is not too startling, but the proof is most enjoyable. ##### Theorem EMHE Every Matrix Has an Eigenvalue Suppose A A is a square matrix. Then A A has at least one eigenvalue. [Proof](http://linear.ups.edu/html/section-EE.html) The proof of [Theorem EMHE](http://linear.ups.edu/html/section-EE.html "Every Matrix Has an Eigenvalue") is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical. We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue. ##### [Example CAEHW](http://linear.ups.edu/html/section-EE.html) Computing an eigenvalue the hard way #### Subsection CEE Computing Eigenvalues and Eigenvectors Fortunately, we need not rely on the procedure of [Theorem EMHE](http://linear.ups.edu/html/section-EE.html "Every Matrix Has an Eigenvalue") each time we need an eigenvalue. It is the determinant, and specifically [Theorem SMZD](http://linear.ups.edu/html/section-EE.html "Singular Matrices have Zero Determinants"), that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, Ax\=λx⟺Ax−λInx\=0⟺(A−λIn)x\=0 A x \= λ x ⟺ A x − λ I n x \= 0 ⟺ ( A − λ I n ) x \= 0 So, for an eigenvalue λ λ and associated eigenvector x≠0 x ≠ 0, the vector x x will be a nonzero element of the null space of A−λIn A − λ I n, while the matrix A−λIn A − λ I n will be singular and therefore have zero determinant. These ideas are made precise in [Theorem EMRCP](http://linear.ups.edu/html/section-EE.html "Eigenvalues of a Matrix are Roots of Characteristic Polynomials") and [Theorem EMNS](http://linear.ups.edu/html/section-EE.html "Eigenspace of a Matrix is a Null Space"), but for now this brief discussion should suffice as motivation for the following definition and example. ##### Definition CP Characteristic Polynomial Suppose that A A is a square matrix of size n n. Then the characteristic polynomial of A A is the polynomial pA(x) p A ( x ) defined by pA(x)\=det(A−xIn) p A ( x ) \= det ( A − x I n ) ##### [Example CPMS3](http://linear.ups.edu/html/section-EE.html) Characteristic polynomial of a matrix, size 3 The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues. ##### Theorem EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials Suppose A A is a square matrix. Then λ λ is an eigenvalue of A A if and only if pA(λ)\=0 p A ( λ ) \= 0. [Proof](http://linear.ups.edu/html/section-EE.html) ##### [Example EMS3](http://linear.ups.edu/html/section-EE.html) Eigenvalues of a matrix, size 3 Let us now turn our attention to the computation of eigenvectors. ##### Definition EM Eigenspace of a Matrix Suppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the eigenspace of A A for λ λ, EA(λ) E A ( λ ), is the set of all the eigenvectors of A A for λ λ, together with the inclusion of the zero vector. [Example SEE](http://linear.ups.edu/html/section-EE.html "Some eigenvalues and eigenvectors") hinted that the set of eigenvectors for a single eigenvalue might have some closure properties, and with the addition of the one eigenvector that is never an eigenvector, 0 0, we indeed get a whole subspace. ##### Theorem EMS Eigenspace for a Matrix is a Subspace Suppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A. Then the eigenspace EA(λ) E A ( λ ) is a subspace of the vector space Cn C n. [Proof](http://linear.ups.edu/html/section-EE.html) [Theorem EMS](http://linear.ups.edu/html/section-EE.html "Eigenspace for a Matrix is a Subspace") tells us that an eigenspace is a subspace (and hence a vector space in its own right). Our next theorem tells us how to quickly construct this subspace. ##### Theorem EMNS Eigenspace of a Matrix is a Null Space Suppose A A is a square matrix of size n n and λ λ is an eigenvalue of A A. Then EA(λ)\=N(A−λIn) E A ( λ ) \= N ( A − λ I n ) [Proof](http://linear.ups.edu/html/section-EE.html) You might notice the close parallels (and differences) between the proofs of [Theorem EMRCP](http://linear.ups.edu/html/section-EE.html "Eigenvalues of a Matrix are Roots of Characteristic Polynomials") and [Theorem EMNS](http://linear.ups.edu/html/section-EE.html "Eigenspace of a Matrix is a Null Space"). Since [Theorem EMNS](http://linear.ups.edu/html/section-EE.html "Eigenspace of a Matrix is a Null Space") describes the set of all the eigenvectors of A A as a null space we can use techniques such as [Theorem BNS](http://linear.ups.edu/html/section-EE.html "Basis for Null Spaces") to provide concise descriptions of eigenspaces. [Theorem EMNS](http://linear.ups.edu/html/section-EE.html "Eigenspace of a Matrix is a Null Space") also provides a trivial proof for [Theorem EMS](http://linear.ups.edu/html/section-EE.html "Eigenspace for a Matrix is a Subspace"). ##### [Example ESMS3](http://linear.ups.edu/html/section-EE.html) Eigenspaces of a matrix, size 3 #### Subsection ECEE Examples of Computing Eigenvalues and Eigenvectors There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of [Example CPMS3](http://linear.ups.edu/html/section-EE.html "Characteristic polynomial of a matrix, size 3"), [Example EMS3](http://linear.ups.edu/html/section-EE.html "Eigenvalues of a matrix, size 3") and [Example ESMS3](http://linear.ups.edu/html/section-EE.html "Eigenspaces of a matrix, size 3"). First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples. ##### Definition AME Algebraic Multiplicity of an Eigenvalue Suppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the algebraic multiplicity of λ λ, αA(λ) α A ( λ ), is the highest power of (x−λ) ( x − λ ) that divides the characteristic polynomial, pA(x) p A ( x ). Since an eigenvalue λ λ is a root of the characteristic polynomial, there is always a factor of (x−λ) ( x − λ ), and the algebraic multiplicity is just the power of this factor in a factorization of pA(x) p A ( x ). So in particular, αA(λ)≥1 α A ( λ ) ≥ 1. Compare the definition of algebraic multiplicity with the next definition. ##### Definition GME Geometric Multiplicity of an Eigenvalue Suppose that A A is a square matrix and λ λ is an eigenvalue of A A. Then the geometric multiplicity of λ λ, γA(λ) γ A ( λ ), is the dimension of the eigenspace EA(λ) E A ( λ ). Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so γA(λ)≥1 γ A ( λ ) ≥ 1. ##### [Example EMMS4](http://linear.ups.edu/html/section-EE.html) Eigenvalue multiplicities, matrix of size 4 ##### [Example ESMS4](http://linear.ups.edu/html/section-EE.html) Eigenvalues, symmetric matrix of size 4 ##### [Example HMEM5](http://linear.ups.edu/html/section-EE.html) High multiplicity eigenvalues, matrix of size 5 ##### [Example CEMS6](http://linear.ups.edu/html/section-EE.html) Complex eigenvalues, matrix of size 6 ##### [Example DEMS5](http://linear.ups.edu/html/section-EE.html) Distinct eigenvalues, matrix of size 5 ##### [Sage CEVAL](http://linear.ups.edu/html/section-EE.html) Computing Eigenvalues ##### [Sage CEVEC](http://linear.ups.edu/html/section-EE.html) Computing Eigenvectors #### [Reading Questions](http://linear.ups.edu/html/section-EE.html) #### [Exercises](http://linear.ups.edu/html/section-EE.html) [![web analytics](http://c.statcounter.com/8375157/0/c03f6ece/1/)](http://statcounter.com/ "web analytics")
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