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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 185 (from laksa161)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa185.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, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/algebra/eigenvalues/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/algebra/eigenvalues/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.cuemath.com\/algebra\/eigenvalues\/","crawl_time":1776846725,"first_indexed_time":1642822718,"http_code":200,"src_unparsed":"com,cuemath!www,\/algebra\/eigenvalues\/ s443","src_root_hash":"6203651625873809585","history_drop_reason":null,"meta_title":"Eigenvalues - Examples | How to Find Eigenvalues of Matrix?","meta_descriptions":["The eigenvalues of a matrix are the scalars by which eigenvectors change when some transformation is applied to them. Learn how to find the eigenvalues of 2x2 and 3x3 matrices using the characteristic equation with examples."],"attrs_boilerpipe_text":"We come up with the terms eigenvalues and eigenvectors when we study the linear transformations. Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix.\nLet us learn more about the eigenvalues of matrix along with their properties and examples.\n1.\nWhat are Eigenvalues of Matrix?\n2.\nHow to Find Eigenvalues?\n3.\nEigenvalues of a 2x2 Matrix\n4.\nEigenvalues of a 3x3 Matrix\n5.\nProperties of Eigenvalues\n6.\nApplications of Eigenvalues\n7.\nFAQs on Eigenvalues\nThe\neigenvalues of matrix\nare scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and\nv\nis a non-zero column vector of order n x 1 such that\nAv\n=\nλv\n(it means that the product of A and\nv\nis just a scalar multiple of\nv\n), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector\nv\n.\nThe word \"eigen\" is from the German language and it means \"characteristic\", \"proper\", or \"own\". Thus, eigenvalues are also known as \"characteristic values\" (or) \"characteristic roots\" (or) \"proper values\", etc. The mathematical definition of eigenvalues is as shown below.\nFrom the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then\nA\nv\n= λ\nv\nIf I is the\nidentity matrix\nof the same order as A, then we can write the above equation as\nA\nv\n= λ (I\nv\n) (because\nv\n= I\nv\n)\nA\nv\n- λ (I\nv\n) = 0\nTaking\nv\nas common factor,\nv\n(A - λI) = 0\nThis represents a\nhomogeneous system of linear equations\nand it has a non-trivial solution only when the determinant of the coefficient matrix is 0.\ni.e.,\n|A - λI| = 0\nThis equation is called the\ncharacteristic equation\n(where |A - λI| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A.\nTake the identity matrix I whose order is the same as A.\nMultiply every element of I by λ to get λI.\nSubtract λI from A to get A - λI.\nFind its\ndeterminant\n.\nSet the determinant to zero and solve for λ.\nLet us apply these steps to find the eigenvalues of matrices of different orders.\nLet us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A =\n[\n5\n4\n1\n2\n]\n. Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I =\n[\n1\n0\n0\n1\n]\n. Then\nλI = λ\n[\n1\n0\n0\n1\n]\n=\n[\nλ\n0\n0\nλ\n]\nA - λI =\n[\n5\n4\n1\n2\n]\n-\n[\nλ\n0\n0\nλ\n]\n=\n[\n5\n−\nλ\n4\n1\n2\n−\nλ\n]\nIts determinant is,\n|A - λI| = (5 - λ) (2 - λ) - (1)(4)\n= 10 - 5λ - 2λ + λ\n2\n- 4\n= λ\n2\n- 7λ + 6\nThe characteristic equation is,\n|A - λI| = 0\nλ\n2\n- 7λ + 6 = 0\n(λ - 6)(λ - 1) = 0\nλ - 6 = 0; λ - 1 = 0\nλ = 6; λ = 1\nThus, the eigenvalues of matrix A are 1 and 6.\nLet us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A =\n[\n3\n1\n1\n2\n4\n2\n1\n1\n3\n]\n.\nThe characteristic equation is,\n|A - λI| = 0\n|\n3\n−\nλ\n1\n1\n2\n4\n−\nλ\n2\n1\n1\n3\n−\nλ\n|\n= 0\n(3 - λ) [(4 - λ)(3 - λ) - 2(1) ] - 1 [ 2(3 - λ) - 2(1) ] + 1 [2 (1) - 1 (4 - λ) ] = 0\n(3 - λ) [12 - 4λ - 3λ + λ\n2\n- 2] - 6 + 2λ + 2 + 2 - 4 + λ = 0\n(3 - λ) [10 - 7λ + λ\n2\n] - 6 + 3λ = 0\n30 - 21λ + 3λ\n2\n- 10λ + 7λ\n2\n- λ\n3\n- 6 + 3λ = 0\n-λ\n3\n+ 10λ\n2\n- 28λ + 24 = 0\nMultiplying both sides by -1,\nλ\n3\n- 10λ\n2\n+ 28λ - 24 = 0\nThis is a\ncubic equation\n. We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using\nsynthetic division\n,\nSet the quotient equal to 0.\nλ\n2\n- 8λ + 12 = 0\n(λ - 6)(λ - 2) = 0\nλ = 6; λ = 2\nThus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6.\nA\nsquare matrix\nof order n has at most n eigenvalues.\nAn identity matrix has only one eigenvalue which is 1.\nThe eigenvalues of triangular matrices and\ndiagonal matrices\nare nothing but the elements of their principal diagonal.\nThe sum of eigenvalues of matrix A is equal to the sum of its diagonal elements.\nThe product of eigenvalues of matrix A is equal to its\ndeterminant\n.\nThe eigenvalues of\nhermitian\nand\nsymmetric\nmatrices are real.\nThe eigenvalues of\nskew hermitian\nand\nskew-symmetric\nmatrices are either zeros are purely\nimaginary numbers\n.\nA matrix and its\ntranspose\nhave the same eigenvalues.\nIf A and B are two square matrices of the same order, then AB and BA have the same eigenvalues.\nThe eigenvalues of an\northogonal matrix\nare 1 and -1.\nIf λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar.\nIf λ is an eigenvalue of A, then λ\nk\nis an eigenvalue of A\nk\n.\nIf λ is an eigenvalue of A, then 1\/λ is an eigenvalue of A\n-1\n(if the\ninverse\nof A exists).\nIf λ is an eigenvalue of A, then |A| \/ λ is an eigenvalue of the\nadjoint\nof A.\nApart from these properties, we have another theorem related to eigenvalues called the \"\nCaley-Hamilton Theorem\n\"\n. It says, \"every square matrix satisfies its characteristic equation\". i.e., if A is a square matrix then it satisfies |A - λI| = 0. For example, if λ\n2\n- 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A\n2\n- 8A + 12 = 0.\nEigenvalues are used in electric circuits, quantum mechanics, control theory, etc.\nThey are used in the design of car stereo systems.\nThey are also used to design bridges.\nIt is not surprising to know that eigenvalues are also used in determining Google's page rank.\nThey are used in geometric transformations.\nRead More","attrs_markdown":"[Math Tutoring](https:\/\/www.cuemath.com\/math-online-classes\/)\n\n[Elementary K-2](https:\/\/www.cuemath.com\/math-online-classes\/grade-k-2\/)\n\n[Kindergarten](https:\/\/www.cuemath.com\/math-online-classes\/kindergarten\/)\n\n[Grade 1](https:\/\/www.cuemath.com\/math-online-classes\/grade-1\/)\n\n[Grade 2](https:\/\/www.cuemath.com\/math-online-classes\/grade-2\/)\n\n[Elementary 3-5](https:\/\/www.cuemath.com\/math-online-classes\/grade-3-5\/)\n\n[Grade 3](https:\/\/www.cuemath.com\/math-online-classes\/grade-3\/)\n\n[Grade 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App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp)\n\n[Learn]()\n\n[Practice]()[Download]()\n\n# Eigenvalues\nWe come up with the terms eigenvalues and eigenvectors when we study the linear transformations. Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix.\n\nLet us learn more about the eigenvalues of matrix along with their properties and examples.\n\n|   |   |\n|---|---|\n| 1\\. | [What are Eigenvalues of Matrix?](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#what-are-eigenvalues-of-matrix) |\n| 2\\. | [How to Find Eigenvalues?](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#how-to-find-eigenvalues) |\n| 3\\. | [Eigenvalues of a 2x2 Matrix](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#eigenvalues-of-a-2x2-matrix) |\n| 4\\. | [Eigenvalues of a 3x3 Matrix](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#eigenvalues-of-a-3x3-matrix) |\n| 5\\. | [Properties of Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#properties-of-eigenvalues) |\n| 6\\. | [Applications of Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#applications) |\n| 7\\. | [FAQs on Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#faqs) |\n## What are Eigenvalues of Matrix?\nThe **eigenvalues of matrix** are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and **v** is a non-zero column vector of order n x 1 such that **Av** = **λv** (it means that the product of A and **v** is just a scalar multiple of **v**), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector **v**.\n\nThe word \"eigen\" is from the German language and it means \"characteristic\", \"proper\", or \"own\". Thus, eigenvalues are also known as \"characteristic values\" (or) \"characteristic roots\" (or) \"proper values\", etc. The mathematical definition of eigenvalues is as shown below.\n\n![eigenvalues definition](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/eigenvalues-of-matrix-definition-1641897493.png)\n\n## How to Find Eigenvalues?\nFrom the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then\n\nA**v** = λ**v**\n\nIf I is the [identity matrix](https:\/\/www.cuemath.com\/algebra\/identity-matrix\/) of the same order as A, then we can write the above equation as\n\nA**v** = λ (I**v**) (because **v** = I**v**)\n\nA**v** - λ (I**v**) = 0\n\nTaking **v** as common factor,\n\n**v** (A - λI) = 0\n\nThis represents a [homogeneous system of linear equations](https:\/\/www.cuemath.com\/algebra\/homogeneous-system-of-linear-equations\/) and it has a non-trivial solution only when the determinant of the coefficient matrix is 0.\n\ni.e., **\\|A - λI\\| = 0**\n\nThis equation is called the **characteristic equation** (where \\|A - λI\\| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A.\n\n1. Take the identity matrix I whose order is the same as A.\n2. Multiply every element of I by λ to get λI.\n3. Subtract λI from A to get A - λI.\n4. Find its [determinant](https:\/\/www.cuemath.com\/algebra\/determinants\/).\n5. Set the determinant to zero and solve for λ.\n\nLet us apply these steps to find the eigenvalues of matrices of different orders.\n\n## Eigenvalues of a 2x2 Matrix\nLet us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = ⎡⎢⎣5412⎤⎥⎦ \\[ 5 4 1 2 \\]. Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = ⎡⎢⎣1001⎤⎥⎦ \\[ 1 0 0 1 \\]. Then\n\nλI = λ ⎡⎢⎣1001⎤⎥⎦ \\[ 1 0 0 1 \\] = ⎡⎢⎣λ00λ⎤⎥⎦ \\[ λ 0 0 λ \\]\n\nA - λI = ⎡⎢⎣5412⎤⎥⎦ \\[ 5 4 1 2 \\] - ⎡⎢⎣λ00λ⎤⎥⎦ \\[ λ 0 0 λ \\]\n\n\\= ⎡⎢⎣5−λ412−λ⎤⎥⎦ \\[ 5 − λ 4 1 2 − λ \\]\n\nIts determinant is,\n\n\\|A - λI\\| = (5 - λ) (2 - λ) - (1)(4)\n\n\\= 10 - 5λ - 2λ + λ2 - 4\n\n\\= λ2 - 7λ + 6\n\nThe characteristic equation is,\n\n\\|A - λI\\| = 0\n\nλ2 - 7λ + 6 = 0\n\n(λ - 6)(λ - 1) = 0\n\nλ - 6 = 0; λ - 1 = 0\n\nλ = 6; λ = 1\n\nThus, the eigenvalues of matrix A are 1 and 6.\n\n## Eigenvalues of a 3x3 Matrix\nLet us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = ⎡⎢⎣311242113⎤⎥⎦ \\[ 3 1 1 2 4 2 1 1 3 \\].\n\nThe characteristic equation is,\n\n\\|A - λI\\| = 0\n\n∣∣ ∣∣3−λ1124−λ2113−λ∣∣ ∣∣ \\| 3 − λ 1 1 2 4 − λ 2 1 1 3 − λ \\| = 0\n\n(3 - λ) \\[(4 - λ)(3 - λ) - 2(1) \\] - 1 \\[ 2(3 - λ) - 2(1) \\] + 1 \\[2 (1) - 1 (4 - λ) \\] = 0\n\n(3 - λ) \\[12 - 4λ - 3λ + λ2 - 2\\] - 6 + 2λ + 2 + 2 - 4 + λ = 0\n\n(3 - λ) \\[10 - 7λ + λ2\\] - 6 + 3λ = 0\n\n30 - 21λ + 3λ2 \\- 10λ + 7λ2 - λ3 - 6 + 3λ = 0\n\n\\-λ3 + 10λ2 \\- 28λ + 24 = 0\n\nMultiplying both sides by -1,\n\nλ3 - 10λ2 \\+ 28λ - 24 = 0\n\nThis is a [cubic equation](https:\/\/www.cuemath.com\/calculators\/cubic-equation-solver\/). We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using [synthetic division](https:\/\/www.cuemath.com\/algebra\/synthetic-division-of-polynomial\/),\n\n![finding eigenvalues of 3x3 matrix using synthetic division](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/eigenvalues-of-3x3-matrix-1641915065.png)\n\nSet the quotient equal to 0.\n\nλ2 - 8λ + 12 = 0\n\n(λ - 6)(λ - 2) = 0\n\nλ = 6; λ = 2\n\nThus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6.\n\n## Properties of Eigenvalues\n- A [square matrix](https:\/\/www.cuemath.com\/algebra\/square-matrix\/) of order n has at most n eigenvalues.\n- An identity matrix has only one eigenvalue which is 1.\n- The eigenvalues of triangular matrices and [diagonal matrices](https:\/\/www.cuemath.com\/algebra\/diagonal-matrix\/) are nothing but the elements of their principal diagonal.\n- The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements.\n- The product of eigenvalues of matrix A is equal to its [determinant](https:\/\/www.cuemath.com\/algebra\/determinant-of-matrix\/).\n- The eigenvalues of [hermitian](https:\/\/www.cuemath.com\/algebra\/hermitian-matrix\/) and [symmetric](https:\/\/www.cuemath.com\/algebra\/symmetric-matrix\/) matrices are real.\n- The eigenvalues of [skew hermitian](https:\/\/www.cuemath.com\/algebra\/skew-hermitian-matrix\/) and [skew-symmetric](https:\/\/www.cuemath.com\/algebra\/skew-symmetric-matrix\/) matrices are either zeros are purely [imaginary numbers](https:\/\/www.cuemath.com\/numbers\/imaginary-numbers\/).\n- A matrix and its [transpose](https:\/\/www.cuemath.com\/algebra\/transpose-of-a-matrix\/) have the same eigenvalues.\n- If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues.\n- The eigenvalues of an [orthogonal matrix](https:\/\/www.cuemath.com\/algebra\/orthogonal-matrix\/) are 1 and -1.\n- If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar.\n- If λ is an eigenvalue of A, then λk is an eigenvalue of Ak.\n- If λ is an eigenvalue of A, then 1\/λ is an eigenvalue of A\\-1 (if the [inverse](https:\/\/www.cuemath.com\/algebra\/inverse-of-a-matrix\/) of A exists).\n- If λ is an eigenvalue of A, then \\|A\\| \/ λ is an eigenvalue of the [adjoint](https:\/\/www.cuemath.com\/algebra\/adjoint-of-a-matrix\/) of A.\n\nApart from these properties, we have another theorem related to eigenvalues called the \"**[Caley-Hamilton Theorem](https:\/\/www.cuemath.com\/algebra\/cayley-hamilton-theorem\/)\"**. It says, \"every square matrix satisfies its characteristic equation\". i.e., if A is a square matrix then it satisfies \\|A - λI\\| = 0. For example, if λ2 - 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A2 - 8A + 12 = 0.\n\n## Applications of Eigenvalues\n- Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc.\n- They are used in the design of car stereo systems.\n- They are also used to design bridges.\n- It is not surprising to know that eigenvalues are also used in determining Google's page rank.\n- They are used in geometric transformations.\n\nRead More\n\nExplore  \n math program\n\nDownload FREE Study Materials\n\n![](data:image\/svg+xml;base64,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)\n\nSHEETS\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/thumbnails\/eigenvalues-worksheet-newui-first-page-1641964487.jpg)\n\nEigenvalues Worksheet\n\nMatrices Worksheet\n\n![](data:image\/svg+xml;base64,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)\n\n## Examples of Eigenvalues\n1. **Example 1:** Find the eigenvalues of the matrix ⎡⎢⎣300−12020−3⎤⎥⎦ \\[ 3 0 0 − 1 2 0 2 0 − 3 \\].\n   \n   **Solution:**\n   \n   **Method 1:**\n   \n   The given matrix is a lower triangular matrix. Hence its eigenvalues are nothing but its diagonal elements which are 3, 2, and -3.\n   \n   **Method 2:**\n   \n   The characteristic equation of the given matrix is:\n   \n   \\|A - λI\\| = 0\n   \n   ∣∣ ∣∣3−λ00−12−λ020−3−λ∣∣ ∣∣ \\| 3 − λ 0 0 − 1 2 − λ 0 2 0 − 3 − λ \\| = 0\n   \n   Expanding the determinant using the first row:\n   \n   (3 - λ)(2 - λ)(-3 - λ) = 0\n   \n   3 - λ = 0; 2 - λ = 0; -3 - λ = 0\n   \n   λ = 3; λ = 2; λ = -3\n   \n   **Answer:** 3, 2, and -3.\n2. **Example 2:** Prove that the eigenvalues of the 2x2 matrix ⎡⎢⎣0−110⎤⎥⎦ \\[ 0 − 1 1 0 \\]\n   \n   **Solution:**\n   \n   The characteristic equation of the given matrix is:\n   \n   \\|A - λI\\| = 0\n   \n   ∣∣ ∣∣0−λ−110−λ∣∣ ∣∣ \\| 0 − λ − 1 1 0 − λ \\| = 0\n   \n   ∣∣ ∣∣−λ−11−λ∣∣ ∣∣ \\| − λ − 1 1 − λ \\| = 0\n   \n   λ2 + 1 = 0\n   \n   λ2 = -1\n   \n   λ = ± √(-1) = ± i\n   \n   But eigenvalues should be [real numbers](https:\/\/www.cuemath.com\/numbers\/real-numbers\/).\n   \n   **Answer:** The given matrix has no eigenvalues.\n3. **Example 3:** Find the eigenvalues of ∣∣ ∣∣111111111∣∣ ∣∣ \\| 1 1 1 1 1 1 1 1 1 \\|.\n   \n   **Solution:**\n   \n   The characteristic equation of the given matrix is:\n   \n   \\|A - λI\\| = 0\n   \n   ⎡⎢⎣1−λ1111−λ1111−λ⎤⎥⎦ \\[ 1 − λ 1 1 1 1 − λ 1 1 1 1 − λ \\]\n   \n   (1 - λ) \\[ (1 - λ)(1 - λ) - 1 \\] - 1 (1 - λ - 1) + 1 (1 - 1 + λ) = 0\n   \n   (1 - λ) (1 - 2λ + λ2 - 1) + λ + λ = 0\n   \n   (1 - λ) (λ2 -2λ) + 2λ = 0\n   \n   λ2 - 2λ - λ3 + 2λ2 + 2λ = 0\n   \n   \\-λ3 + 3λ2 = 0\n   \n   λ2 (-λ + 3) = 0\n   \n   λ2 = 0; -λ + 3 = 0\n   \n   λ = 0, 0, 3\n   \n   **Answer:** 0, 0, 3.\n\nView Answer \\>\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/caucasian-kid-and-indian-teacher-grade-4-7-1619081163.png)\n\nBreakdown tough concepts through simple visuals.\n\nMath will no longer be a tough subject, especially when you understand the concepts through visualizations.\n\n[Book a Free Trial Class](https:\/\/www.cuemath.com\/parent\/signup\/)\n\n## Practice Questions on Eigenvalues\nCheck Answer \\>\n\n## FAQs on Eigenvalues\n### What is the Definition of Eigenvalues?\n**Eigenvalues** of a matrix are scalars by which eigenvectors change when the matrix or transformation is applied to it. Mathematically, if A**v** = λ**v**, then\n\n- λ is called the eigenvalue\n- **v** is called the corresponding eigenvector\n\n### How can We Find the Eigenvalues of Matrix?\nTo find the eigenvalues of a square matrix A:\n\n- Find its characteristic equation using \\|A - λI\\| = 0, where I is the identity matrix of same order A.\n- Solve it for λ and the solutions would give the eigenvalues.\n\n### What are the Eigenvalues of a Diagonal Matrix?\nWe know that all the elements of a diagonal matrix other than its diagonal elements are zeros. Hence, the eigen values of a diagonal matrix are just its diagonal elements.\n\n### How to Find Eigenvalues and Eigenvectors?\nFor any square matrix A:\n\n- Solve \\|A - λI\\| = 0 for λ to find eigenvalues.\n- Solve (A - λI) **v** = 0 for **v** to get corresponding eigenvectors.\n\n### Where Can We Find Eigenvalue Calculator?\nWe can find the eigenvalue calculator by clicking [here](https:\/\/www.cuemath.com\/calculators\/eigenvalue-calculator\/). Here, you can enter any 2x2 matrix, then it will show you the eigenvalues along with steps.\n\n### What is Characteristic Equation For Finding Eigenvalues?\nIf A is a square matrix and λ represents its eigenvalues then \\|A - λI\\| = 0 represents its characteristic equation and by solving this would result in the eigenvalues.\n\n### What are the Eigenvalues of an Upper Triangular Matrix?\nSince in [upper triangular matrix](https:\/\/www.cuemath.com\/algebra\/triangular-matrix\/), all elements under the principal diagonal are zeros, the eigenvalues are nothing but the diagonal elements of the matrix.\n\n### What are the Eigenvalues of a Unitary Matrix?\nA [unitary matrix](https:\/\/www.cuemath.com\/algebra\/unitary-matrix\/) is a complex matrix such that its inverse is equal to its conjugate transpose. 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Sector 42, Gurugram, Haryana 122009\n\nUS Office\n\nCueLearn Inc, 1013 Centre Road, Suite 403-B, Wilmington, Delaware 19805\n\nIndia Office\n\nPlot No. F-17\/5, Golf Course Rd, Sector 42, Gurugram, Haryana 122009\n\n![Cuemath Logo](https:\/\/d138zd1ktt9iqe.cloudfront.net\/static\/website-v3\/cuemath-logo.svg)\n\n[Terms & conditions](https:\/\/www.cuemath.com\/terms)\n\n[Privacy Policy](https:\/\/www.cuemath.com\/privacy)\n\n[![Facebook](https:\/\/d138zd1ktt9iqe.cloudfront.net\/static\/website-v3\/social-media-icons\/facebook-grey.svg)![Facebook hover](https:\/\/d138zd1ktt9iqe.cloudfront.net\/static\/website-v3\/social-media-icons\/facebook.svg)](https:\/\/www.facebook.com\/cuemath\/)[![Youtube](https:\/\/d138zd1ktt9iqe.cloudfront.net\/static\/website-v3\/social-media-icons\/youtube-grey.svg)![Youtube 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Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix.\n\nLet us learn more about the eigenvalues of matrix along with their properties and examples.\n\n|   |   |\n|---|---|\n| 1\\. | [What are Eigenvalues of Matrix?](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#what-are-eigenvalues-of-matrix) |\n| 2\\. | [How to Find Eigenvalues?](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#how-to-find-eigenvalues) |\n| 3\\. | [Eigenvalues of a 2x2 Matrix](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#eigenvalues-of-a-2x2-matrix) |\n| 4\\. | [Eigenvalues of a 3x3 Matrix](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#eigenvalues-of-a-3x3-matrix) |\n| 5\\. | [Properties of Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#properties-of-eigenvalues) |\n| 6\\. | [Applications of Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#applications) |\n| 7\\. | [FAQs on Eigenvalues](https:\/\/www.cuemath.com\/algebra\/eigenvalues\/#faqs) |\n\nThe **eigenvalues of matrix** are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and **v** is a non-zero column vector of order n x 1 such that **Av** = **λv** (it means that the product of A and **v** is just a scalar multiple of **v**), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector **v**.\n\nThe word \"eigen\" is from the German language and it means \"characteristic\", \"proper\", or \"own\". Thus, eigenvalues are also known as \"characteristic values\" (or) \"characteristic roots\" (or) \"proper values\", etc. The mathematical definition of eigenvalues is as shown below.\n\nFrom the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then\n\nA**v** = λ**v**\n\nIf I is the [identity matrix](https:\/\/www.cuemath.com\/algebra\/identity-matrix\/) of the same order as A, then we can write the above equation as\n\nA**v** = λ (I**v**) (because **v** = I**v**)\n\nA**v** - λ (I**v**) = 0\n\nTaking **v** as common factor,\n\n**v** (A - λI) = 0\n\nThis represents a [homogeneous system of linear equations](https:\/\/www.cuemath.com\/algebra\/homogeneous-system-of-linear-equations\/) and it has a non-trivial solution only when the determinant of the coefficient matrix is 0.\n\ni.e., **\\|A - λI\\| = 0**\n\nThis equation is called the **characteristic equation** (where \\|A - λI\\| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A.\n\n1. Take the identity matrix I whose order is the same as A.\n2. Multiply every element of I by λ to get λI.\n3. Subtract λI from A to get A - λI.\n4. Find its [determinant](https:\/\/www.cuemath.com\/algebra\/determinants\/).\n5. Set the determinant to zero and solve for λ.\n\nLet us apply these steps to find the eigenvalues of matrices of different orders.\n\nLet us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = \\[ 5 4 1 2 \\]. Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = \\[ 1 0 0 1 \\]. Then\n\nλI = λ \\[ 1 0 0 1 \\] = \\[ λ 0 0 λ \\]\n\nA - λI = \\[ 5 4 1 2 \\] - \\[ λ 0 0 λ \\]\n\n\\= \\[ 5 − λ 4 1 2 − λ \\]\n\nIts determinant is,\n\n\\|A - λI\\| = (5 - λ) (2 - λ) - (1)(4)\n\n\\= 10 - 5λ - 2λ + λ2 - 4\n\n\\= λ2 - 7λ + 6\n\nThe characteristic equation is,\n\n\\|A - λI\\| = 0\n\nλ2 - 7λ + 6 = 0\n\n(λ - 6)(λ - 1) = 0\n\nλ - 6 = 0; λ - 1 = 0\n\nλ = 6; λ = 1\n\nThus, the eigenvalues of matrix A are 1 and 6.\n\nLet us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = \\[ 3 1 1 2 4 2 1 1 3 \\].\n\nThe characteristic equation is,\n\n\\|A - λI\\| = 0\n\n\\| 3 − λ 1 1 2 4 − λ 2 1 1 3 − λ \\| = 0\n\n(3 - λ) \\[(4 - λ)(3 - λ) - 2(1) \\] - 1 \\[ 2(3 - λ) - 2(1) \\] + 1 \\[2 (1) - 1 (4 - λ) \\] = 0\n\n(3 - λ) \\[12 - 4λ - 3λ + λ2 - 2\\] - 6 + 2λ + 2 + 2 - 4 + λ = 0\n\n(3 - λ) \\[10 - 7λ + λ2\\] - 6 + 3λ = 0\n\n30 - 21λ + 3λ2 \\- 10λ + 7λ2 - λ3 - 6 + 3λ = 0\n\n\\-λ3 + 10λ2 \\- 28λ + 24 = 0\n\nMultiplying both sides by -1,\n\nλ3 - 10λ2 \\+ 28λ - 24 = 0\n\nThis is a [cubic equation](https:\/\/www.cuemath.com\/calculators\/cubic-equation-solver\/). We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using [synthetic division](https:\/\/www.cuemath.com\/algebra\/synthetic-division-of-polynomial\/),\n\nSet the quotient equal to 0.\n\nλ2 - 8λ + 12 = 0\n\n(λ - 6)(λ - 2) = 0\n\nλ = 6; λ = 2\n\nThus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6.\n\n- A [square matrix](https:\/\/www.cuemath.com\/algebra\/square-matrix\/) of order n has at most n eigenvalues.\n- An identity matrix has only one eigenvalue which is 1.\n- The eigenvalues of triangular matrices and [diagonal matrices](https:\/\/www.cuemath.com\/algebra\/diagonal-matrix\/) are nothing but the elements of their principal diagonal.\n- The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements.\n- The product of eigenvalues of matrix A is equal to its [determinant](https:\/\/www.cuemath.com\/algebra\/determinant-of-matrix\/).\n- The eigenvalues of [hermitian](https:\/\/www.cuemath.com\/algebra\/hermitian-matrix\/) and [symmetric](https:\/\/www.cuemath.com\/algebra\/symmetric-matrix\/) matrices are real.\n- The eigenvalues of [skew hermitian](https:\/\/www.cuemath.com\/algebra\/skew-hermitian-matrix\/) and [skew-symmetric](https:\/\/www.cuemath.com\/algebra\/skew-symmetric-matrix\/) matrices are either zeros are purely [imaginary numbers](https:\/\/www.cuemath.com\/numbers\/imaginary-numbers\/).\n- A matrix and its [transpose](https:\/\/www.cuemath.com\/algebra\/transpose-of-a-matrix\/) have the same eigenvalues.\n- If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues.\n- The eigenvalues of an [orthogonal matrix](https:\/\/www.cuemath.com\/algebra\/orthogonal-matrix\/) are 1 and -1.\n- If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar.\n- If λ is an eigenvalue of A, then λk is an eigenvalue of Ak.\n- If λ is an eigenvalue of A, then 1\/λ is an eigenvalue of A\\-1 (if the [inverse](https:\/\/www.cuemath.com\/algebra\/inverse-of-a-matrix\/) of A exists).\n- If λ is an eigenvalue of A, then \\|A\\| \/ λ is an eigenvalue of the [adjoint](https:\/\/www.cuemath.com\/algebra\/adjoint-of-a-matrix\/) of A.\n\nApart from these properties, we have another theorem related to eigenvalues called the \"**[Caley-Hamilton Theorem](https:\/\/www.cuemath.com\/algebra\/cayley-hamilton-theorem\/)\"**. It says, \"every square matrix satisfies its characteristic equation\". i.e., if A is a square matrix then it satisfies \\|A - λI\\| = 0. For example, if λ2 - 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A2 - 8A + 12 = 0.\n\n- Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc.\n- They are used in the design of car stereo systems.\n- They are also used to design bridges.\n- It is not surprising to know that eigenvalues are also used in determining Google's page rank.\n- They are used in geometric transformations.\n\nRead More","meta_canonical":null,"ml_categories_json":"{\"\/Science\":891,\"\/Science\/Mathematics\":881,\"\/Science\/Mathematics\/Other\":809}","ml_types_json":"{\"\/Article\":998,\"\/Article\/Tutorial_or_Guide\":835}","ml_intent_types_json":"{\"Informational\":999}","meta_language":"en","attrs_author":null,"attrs_publish_time":0,"attrs_original_publish_time":1642822718,"attrs_is_republished":0,"attrs_nr_words":"2987","attrs_boilerpipe_nr_words":"1242","body_ext_links_number":10,"body_int_links_number":131,"meta_nofollow":0,"meta_noarchive":0,"props_was_rendered":1,"src_redirect":"","download_time_msec":1097,"download_ttfb_msec":942,"download_size":48107}

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Eigenvalues - Examples | How to Find Eigenvalues of Matrix?

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The eigenvalues of a matrix are the scalars by which eigenvectors change when some transformation is applied to them. Learn how to find the eigenvalues of 2x2 and 3x3 matrices using the characteristic equation with examples.

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We come up with the terms eigenvalues and eigenvectors when we study the linear transformations. Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix. Let us learn more about the eigenvalues of matrix along with their properties and examples. 1. What are Eigenvalues of Matrix? 2. How to Find Eigenvalues? 3. Eigenvalues of a 2x2 Matrix 4. Eigenvalues of a 3x3 Matrix 5. Properties of Eigenvalues 6. Applications of Eigenvalues 7. FAQs on Eigenvalues The eigenvalues of matrix are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and v is a non-zero column vector of order n x 1 such that Av = λv (it means that the product of A and v is just a scalar multiple of v ), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector v . The word "eigen" is from the German language and it means "characteristic", "proper", or "own". Thus, eigenvalues are also known as "characteristic values" (or) "characteristic roots" (or) "proper values", etc. The mathematical definition of eigenvalues is as shown below. From the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then A v = λ v If I is the identity matrix of the same order as A, then we can write the above equation as A v = λ (I v ) (because v = I v ) A v - λ (I v ) = 0 Taking v as common factor, v (A - λI) = 0 This represents a homogeneous system of linear equations and it has a non-trivial solution only when the determinant of the coefficient matrix is 0. i.e., |A - λI| = 0 This equation is called the characteristic equation (where |A - λI| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A. Take the identity matrix I whose order is the same as A. Multiply every element of I by λ to get λI. Subtract λI from A to get A - λI. Find its determinant . Set the determinant to zero and solve for λ. Let us apply these steps to find the eigenvalues of matrices of different orders. Let us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = [ 5 4 1 2 ] . Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = [ 1 0 0 1 ] . Then λI = λ [ 1 0 0 1 ] = [ λ 0 0 λ ] A - λI = [ 5 4 1 2 ] - [ λ 0 0 λ ] = [ 5 − λ 4 1 2 − λ ] Its determinant is, |A - λI| = (5 - λ) (2 - λ) - (1)(4) = 10 - 5λ - 2λ + λ 2 - 4 = λ 2 - 7λ + 6 The characteristic equation is, |A - λI| = 0 λ 2 - 7λ + 6 = 0 (λ - 6)(λ - 1) = 0 λ - 6 = 0; λ - 1 = 0 λ = 6; λ = 1 Thus, the eigenvalues of matrix A are 1 and 6. Let us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = [ 3 1 1 2 4 2 1 1 3 ] . The characteristic equation is, |A - λI| = 0 | 3 − λ 1 1 2 4 − λ 2 1 1 3 − λ | = 0 (3 - λ) [(4 - λ)(3 - λ) - 2(1) ] - 1 [ 2(3 - λ) - 2(1) ] + 1 [2 (1) - 1 (4 - λ) ] = 0 (3 - λ) [12 - 4λ - 3λ + λ 2 - 2] - 6 + 2λ + 2 + 2 - 4 + λ = 0 (3 - λ) [10 - 7λ + λ 2 ] - 6 + 3λ = 0 30 - 21λ + 3λ 2 - 10λ + 7λ 2 - λ 3 - 6 + 3λ = 0 -λ 3 + 10λ 2 - 28λ + 24 = 0 Multiplying both sides by -1, λ 3 - 10λ 2 + 28λ - 24 = 0 This is a cubic equation . We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using synthetic division , Set the quotient equal to 0. λ 2 - 8λ + 12 = 0 (λ - 6)(λ - 2) = 0 λ = 6; λ = 2 Thus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6. A square matrix of order n has at most n eigenvalues. An identity matrix has only one eigenvalue which is 1. The eigenvalues of triangular matrices and diagonal matrices are nothing but the elements of their principal diagonal. The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements. The product of eigenvalues of matrix A is equal to its determinant . The eigenvalues of hermitian and symmetric matrices are real. The eigenvalues of skew hermitian and skew-symmetric matrices are either zeros are purely imaginary numbers . A matrix and its transpose have the same eigenvalues. If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues. The eigenvalues of an orthogonal matrix are 1 and -1. If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar. If λ is an eigenvalue of A, then λ k is an eigenvalue of A k . If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A -1 (if the inverse of A exists). If λ is an eigenvalue of A, then |A| / λ is an eigenvalue of the adjoint of A. Apart from these properties, we have another theorem related to eigenvalues called the " Caley-Hamilton Theorem " . It says, "every square matrix satisfies its characteristic equation". i.e., if A is a square matrix then it satisfies |A - λI| = 0. For example, if λ 2 - 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A 2 - 8A + 12 = 0. Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc. They are used in the design of car stereo systems. They are also used to design bridges. It is not surprising to know that eigenvalues are also used in determining Google's page rank. They are used in geometric transformations. Read More

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Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix. Let us learn more about the eigenvalues of matrix along with their properties and examples. | | | |---|---| | 1\. | [What are Eigenvalues of Matrix?](https://www.cuemath.com/algebra/eigenvalues/#what-are-eigenvalues-of-matrix) | | 2\. | [How to Find Eigenvalues?](https://www.cuemath.com/algebra/eigenvalues/#how-to-find-eigenvalues) | | 3\. | [Eigenvalues of a 2x2 Matrix](https://www.cuemath.com/algebra/eigenvalues/#eigenvalues-of-a-2x2-matrix) | | 4\. | [Eigenvalues of a 3x3 Matrix](https://www.cuemath.com/algebra/eigenvalues/#eigenvalues-of-a-3x3-matrix) | | 5\. | [Properties of Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#properties-of-eigenvalues) | | 6\. | [Applications of Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#applications) | | 7\. | [FAQs on Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#faqs) | ## What are Eigenvalues of Matrix? The **eigenvalues of matrix** are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and **v** is a non-zero column vector of order n x 1 such that **Av** = **λv** (it means that the product of A and **v** is just a scalar multiple of **v**), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector **v**. The word "eigen" is from the German language and it means "characteristic", "proper", or "own". Thus, eigenvalues are also known as "characteristic values" (or) "characteristic roots" (or) "proper values", etc. The mathematical definition of eigenvalues is as shown below. ![eigenvalues definition](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/eigenvalues-of-matrix-definition-1641897493.png) ## How to Find Eigenvalues? From the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then A**v** = λ**v** If I is the [identity matrix](https://www.cuemath.com/algebra/identity-matrix/) of the same order as A, then we can write the above equation as A**v** = λ (I**v**) (because **v** = I**v**) A**v** - λ (I**v**) = 0 Taking **v** as common factor, **v** (A - λI) = 0 This represents a [homogeneous system of linear equations](https://www.cuemath.com/algebra/homogeneous-system-of-linear-equations/) and it has a non-trivial solution only when the determinant of the coefficient matrix is 0. i.e., **\|A - λI\| = 0** This equation is called the **characteristic equation** (where \|A - λI\| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A. 1. Take the identity matrix I whose order is the same as A. 2. Multiply every element of I by λ to get λI. 3. Subtract λI from A to get A - λI. 4. Find its [determinant](https://www.cuemath.com/algebra/determinants/). 5. Set the determinant to zero and solve for λ. Let us apply these steps to find the eigenvalues of matrices of different orders. ## Eigenvalues of a 2x2 Matrix Let us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = ⎡⎢⎣5412⎤⎥⎦ \[ 5 4 1 2 \]. Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = ⎡⎢⎣1001⎤⎥⎦ \[ 1 0 0 1 \]. Then λI = λ ⎡⎢⎣1001⎤⎥⎦ \[ 1 0 0 1 \] = ⎡⎢⎣λ00λ⎤⎥⎦ \[ λ 0 0 λ \] A - λI = ⎡⎢⎣5412⎤⎥⎦ \[ 5 4 1 2 \] - ⎡⎢⎣λ00λ⎤⎥⎦ \[ λ 0 0 λ \] \= ⎡⎢⎣5−λ412−λ⎤⎥⎦ \[ 5 − λ 4 1 2 − λ \] Its determinant is, \|A - λI\| = (5 - λ) (2 - λ) - (1)(4) \= 10 - 5λ - 2λ + λ2 - 4 \= λ2 - 7λ + 6 The characteristic equation is, \|A - λI\| = 0 λ2 - 7λ + 6 = 0 (λ - 6)(λ - 1) = 0 λ - 6 = 0; λ - 1 = 0 λ = 6; λ = 1 Thus, the eigenvalues of matrix A are 1 and 6. ## Eigenvalues of a 3x3 Matrix Let us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = ⎡⎢⎣311242113⎤⎥⎦ \[ 3 1 1 2 4 2 1 1 3 \]. The characteristic equation is, \|A - λI\| = 0 ∣∣ ∣∣3−λ1124−λ2113−λ∣∣ ∣∣ \| 3 − λ 1 1 2 4 − λ 2 1 1 3 − λ \| = 0 (3 - λ) \[(4 - λ)(3 - λ) - 2(1) \] - 1 \[ 2(3 - λ) - 2(1) \] + 1 \[2 (1) - 1 (4 - λ) \] = 0 (3 - λ) \[12 - 4λ - 3λ + λ2 - 2\] - 6 + 2λ + 2 + 2 - 4 + λ = 0 (3 - λ) \[10 - 7λ + λ2\] - 6 + 3λ = 0 30 - 21λ + 3λ2 \- 10λ + 7λ2 - λ3 - 6 + 3λ = 0 \-λ3 + 10λ2 \- 28λ + 24 = 0 Multiplying both sides by -1, λ3 - 10λ2 \+ 28λ - 24 = 0 This is a [cubic equation](https://www.cuemath.com/calculators/cubic-equation-solver/). We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using [synthetic division](https://www.cuemath.com/algebra/synthetic-division-of-polynomial/), ![finding eigenvalues of 3x3 matrix using synthetic division](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/eigenvalues-of-3x3-matrix-1641915065.png) Set the quotient equal to 0. λ2 - 8λ + 12 = 0 (λ - 6)(λ - 2) = 0 λ = 6; λ = 2 Thus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6. ## Properties of Eigenvalues - A [square matrix](https://www.cuemath.com/algebra/square-matrix/) of order n has at most n eigenvalues. - An identity matrix has only one eigenvalue which is 1. - The eigenvalues of triangular matrices and [diagonal matrices](https://www.cuemath.com/algebra/diagonal-matrix/) are nothing but the elements of their principal diagonal. - The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements. - The product of eigenvalues of matrix A is equal to its [determinant](https://www.cuemath.com/algebra/determinant-of-matrix/). - The eigenvalues of [hermitian](https://www.cuemath.com/algebra/hermitian-matrix/) and [symmetric](https://www.cuemath.com/algebra/symmetric-matrix/) matrices are real. - The eigenvalues of [skew hermitian](https://www.cuemath.com/algebra/skew-hermitian-matrix/) and [skew-symmetric](https://www.cuemath.com/algebra/skew-symmetric-matrix/) matrices are either zeros are purely [imaginary numbers](https://www.cuemath.com/numbers/imaginary-numbers/). - A matrix and its [transpose](https://www.cuemath.com/algebra/transpose-of-a-matrix/) have the same eigenvalues. - If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues. - The eigenvalues of an [orthogonal matrix](https://www.cuemath.com/algebra/orthogonal-matrix/) are 1 and -1. - If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar. - If λ is an eigenvalue of A, then λk is an eigenvalue of Ak. - If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A\-1 (if the [inverse](https://www.cuemath.com/algebra/inverse-of-a-matrix/) of A exists). - If λ is an eigenvalue of A, then \|A\| / λ is an eigenvalue of the [adjoint](https://www.cuemath.com/algebra/adjoint-of-a-matrix/) of A. Apart from these properties, we have another theorem related to eigenvalues called the "**[Caley-Hamilton Theorem](https://www.cuemath.com/algebra/cayley-hamilton-theorem/)"**. It says, "every square matrix satisfies its characteristic equation". i.e., if A is a square matrix then it satisfies \|A - λI\| = 0. For example, if λ2 - 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A2 - 8A + 12 = 0. ## Applications of Eigenvalues - Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc. - They are used in the design of car stereo systems. - They are also used to design bridges. - It is not surprising to know that eigenvalues are also used in determining Google's page rank. - They are used in geometric transformations. Read More Explore math program Download FREE Study Materials ![](data:image/svg+xml;base64,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) SHEETS ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/thumbnails/eigenvalues-worksheet-newui-first-page-1641964487.jpg) Eigenvalues Worksheet Matrices Worksheet ![](data:image/svg+xml;base64,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) ## Examples of Eigenvalues 1. **Example 1:** Find the eigenvalues of the matrix ⎡⎢⎣300−12020−3⎤⎥⎦ \[ 3 0 0 − 1 2 0 2 0 − 3 \]. **Solution:** **Method 1:** The given matrix is a lower triangular matrix. Hence its eigenvalues are nothing but its diagonal elements which are 3, 2, and -3. **Method 2:** The characteristic equation of the given matrix is: \|A - λI\| = 0 ∣∣ ∣∣3−λ00−12−λ020−3−λ∣∣ ∣∣ \| 3 − λ 0 0 − 1 2 − λ 0 2 0 − 3 − λ \| = 0 Expanding the determinant using the first row: (3 - λ)(2 - λ)(-3 - λ) = 0 3 - λ = 0; 2 - λ = 0; -3 - λ = 0 λ = 3; λ = 2; λ = -3 **Answer:** 3, 2, and -3. 2. **Example 2:** Prove that the eigenvalues of the 2x2 matrix ⎡⎢⎣0−110⎤⎥⎦ \[ 0 − 1 1 0 \] **Solution:** The characteristic equation of the given matrix is: \|A - λI\| = 0 ∣∣ ∣∣0−λ−110−λ∣∣ ∣∣ \| 0 − λ − 1 1 0 − λ \| = 0 ∣∣ ∣∣−λ−11−λ∣∣ ∣∣ \| − λ − 1 1 − λ \| = 0 λ2 + 1 = 0 λ2 = -1 λ = ± √(-1) = ± i But eigenvalues should be [real numbers](https://www.cuemath.com/numbers/real-numbers/). **Answer:** The given matrix has no eigenvalues. 3. **Example 3:** Find the eigenvalues of ∣∣ ∣∣111111111∣∣ ∣∣ \| 1 1 1 1 1 1 1 1 1 \|. **Solution:** The characteristic equation of the given matrix is: \|A - λI\| = 0 ⎡⎢⎣1−λ1111−λ1111−λ⎤⎥⎦ \[ 1 − λ 1 1 1 1 − λ 1 1 1 1 − λ \] (1 - λ) \[ (1 - λ)(1 - λ) - 1 \] - 1 (1 - λ - 1) + 1 (1 - 1 + λ) = 0 (1 - λ) (1 - 2λ + λ2 - 1) + λ + λ = 0 (1 - λ) (λ2 -2λ) + 2λ = 0 λ2 - 2λ - λ3 + 2λ2 + 2λ = 0 \-λ3 + 3λ2 = 0 λ2 (-λ + 3) = 0 λ2 = 0; -λ + 3 = 0 λ = 0, 0, 3 **Answer:** 0, 0, 3. View Answer \> ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/caucasian-kid-and-indian-teacher-grade-4-7-1619081163.png) Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. [Book a Free Trial Class](https://www.cuemath.com/parent/signup/) ## Practice Questions on Eigenvalues Check Answer \> ## FAQs on Eigenvalues ### What is the Definition of Eigenvalues? **Eigenvalues** of a matrix are scalars by which eigenvectors change when the matrix or transformation is applied to it. Mathematically, if A**v** = λ**v**, then - λ is called the eigenvalue - **v** is called the corresponding eigenvector ### How can We Find the Eigenvalues of Matrix? To find the eigenvalues of a square matrix A: - Find its characteristic equation using \|A - λI\| = 0, where I is the identity matrix of same order A. - Solve it for λ and the solutions would give the eigenvalues. ### What are the Eigenvalues of a Diagonal Matrix? We know that all the elements of a diagonal matrix other than its diagonal elements are zeros. Hence, the eigen values of a diagonal matrix are just its diagonal elements. ### How to Find Eigenvalues and Eigenvectors? For any square matrix A: - Solve \|A - λI\| = 0 for λ to find eigenvalues. - Solve (A - λI) **v** = 0 for **v** to get corresponding eigenvectors. ### Where Can We Find Eigenvalue Calculator? We can find the eigenvalue calculator by clicking [here](https://www.cuemath.com/calculators/eigenvalue-calculator/). Here, you can enter any 2x2 matrix, then it will show you the eigenvalues along with steps. ### What is Characteristic Equation For Finding Eigenvalues? If A is a square matrix and λ represents its eigenvalues then \|A - λI\| = 0 represents its characteristic equation and by solving this would result in the eigenvalues. ### What are the Eigenvalues of an Upper Triangular Matrix? Since in [upper triangular matrix](https://www.cuemath.com/algebra/triangular-matrix/), all elements under the principal diagonal are zeros, the eigenvalues are nothing but the diagonal elements of the matrix. ### What are the Eigenvalues of a Unitary Matrix? A [unitary matrix](https://www.cuemath.com/algebra/unitary-matrix/) is a complex matrix such that its inverse is equal to its conjugate transpose. The eigenvalues of a unitary matrix are -1 and 1. Explore math program Math worksheets and visual curriculum Download App Become MathFit™: Boost math skills with daily fun challenges and puzzles. Download the app STRATEGY GAMES LOGIC PUZZLES MENTAL MATH Become MathFit™: Boost math skills with daily fun challenges and puzzles. 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We come up with the terms eigenvalues and eigenvectors when we study the linear transformations. Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix. Let us learn more about the eigenvalues of matrix along with their properties and examples. | | | |---|---| | 1\. | [What are Eigenvalues of Matrix?](https://www.cuemath.com/algebra/eigenvalues/#what-are-eigenvalues-of-matrix) | | 2\. | [How to Find Eigenvalues?](https://www.cuemath.com/algebra/eigenvalues/#how-to-find-eigenvalues) | | 3\. | [Eigenvalues of a 2x2 Matrix](https://www.cuemath.com/algebra/eigenvalues/#eigenvalues-of-a-2x2-matrix) | | 4\. | [Eigenvalues of a 3x3 Matrix](https://www.cuemath.com/algebra/eigenvalues/#eigenvalues-of-a-3x3-matrix) | | 5\. | [Properties of Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#properties-of-eigenvalues) | | 6\. | [Applications of Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#applications) | | 7\. | [FAQs on Eigenvalues](https://www.cuemath.com/algebra/eigenvalues/#faqs) | The **eigenvalues of matrix** are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and **v** is a non-zero column vector of order n x 1 such that **Av** = **λv** (it means that the product of A and **v** is just a scalar multiple of **v**), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector **v**. The word "eigen" is from the German language and it means "characteristic", "proper", or "own". Thus, eigenvalues are also known as "characteristic values" (or) "characteristic roots" (or) "proper values", etc. The mathematical definition of eigenvalues is as shown below. From the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then A**v** = λ**v** If I is the [identity matrix](https://www.cuemath.com/algebra/identity-matrix/) of the same order as A, then we can write the above equation as A**v** = λ (I**v**) (because **v** = I**v**) A**v** - λ (I**v**) = 0 Taking **v** as common factor, **v** (A - λI) = 0 This represents a [homogeneous system of linear equations](https://www.cuemath.com/algebra/homogeneous-system-of-linear-equations/) and it has a non-trivial solution only when the determinant of the coefficient matrix is 0. i.e., **\|A - λI\| = 0** This equation is called the **characteristic equation** (where \|A - λI\| is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the step-by-step process used to find the eigenvalues of a square matrix A. 1. Take the identity matrix I whose order is the same as A. 2. Multiply every element of I by λ to get λI. 3. Subtract λI from A to get A - λI. 4. Find its [determinant](https://www.cuemath.com/algebra/determinants/). 5. Set the determinant to zero and solve for λ. Let us apply these steps to find the eigenvalues of matrices of different orders. Let us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = \[ 5 4 1 2 \]. Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = \[ 1 0 0 1 \]. Then λI = λ \[ 1 0 0 1 \] = \[ λ 0 0 λ \] A - λI = \[ 5 4 1 2 \] - \[ λ 0 0 λ \] \= \[ 5 − λ 4 1 2 − λ \] Its determinant is, \|A - λI\| = (5 - λ) (2 - λ) - (1)(4) \= 10 - 5λ - 2λ + λ2 - 4 \= λ2 - 7λ + 6 The characteristic equation is, \|A - λI\| = 0 λ2 - 7λ + 6 = 0 (λ - 6)(λ - 1) = 0 λ - 6 = 0; λ - 1 = 0 λ = 6; λ = 1 Thus, the eigenvalues of matrix A are 1 and 6. Let us just observe the result of A - λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = \[ 3 1 1 2 4 2 1 1 3 \]. The characteristic equation is, \|A - λI\| = 0 \| 3 − λ 1 1 2 4 − λ 2 1 1 3 − λ \| = 0 (3 - λ) \[(4 - λ)(3 - λ) - 2(1) \] - 1 \[ 2(3 - λ) - 2(1) \] + 1 \[2 (1) - 1 (4 - λ) \] = 0 (3 - λ) \[12 - 4λ - 3λ + λ2 - 2\] - 6 + 2λ + 2 + 2 - 4 + λ = 0 (3 - λ) \[10 - 7λ + λ2\] - 6 + 3λ = 0 30 - 21λ + 3λ2 \- 10λ + 7λ2 - λ3 - 6 + 3λ = 0 \-λ3 + 10λ2 \- 28λ + 24 = 0 Multiplying both sides by -1, λ3 - 10λ2 \+ 28λ - 24 = 0 This is a [cubic equation](https://www.cuemath.com/calculators/cubic-equation-solver/). We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using [synthetic division](https://www.cuemath.com/algebra/synthetic-division-of-polynomial/), Set the quotient equal to 0. λ2 - 8λ + 12 = 0 (λ - 6)(λ - 2) = 0 λ = 6; λ = 2 Thus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6. - A [square matrix](https://www.cuemath.com/algebra/square-matrix/) of order n has at most n eigenvalues. - An identity matrix has only one eigenvalue which is 1. - The eigenvalues of triangular matrices and [diagonal matrices](https://www.cuemath.com/algebra/diagonal-matrix/) are nothing but the elements of their principal diagonal. - The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements. - The product of eigenvalues of matrix A is equal to its [determinant](https://www.cuemath.com/algebra/determinant-of-matrix/). - The eigenvalues of [hermitian](https://www.cuemath.com/algebra/hermitian-matrix/) and [symmetric](https://www.cuemath.com/algebra/symmetric-matrix/) matrices are real. - The eigenvalues of [skew hermitian](https://www.cuemath.com/algebra/skew-hermitian-matrix/) and [skew-symmetric](https://www.cuemath.com/algebra/skew-symmetric-matrix/) matrices are either zeros are purely [imaginary numbers](https://www.cuemath.com/numbers/imaginary-numbers/). - A matrix and its [transpose](https://www.cuemath.com/algebra/transpose-of-a-matrix/) have the same eigenvalues. - If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues. - The eigenvalues of an [orthogonal matrix](https://www.cuemath.com/algebra/orthogonal-matrix/) are 1 and -1. - If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar. - If λ is an eigenvalue of A, then λk is an eigenvalue of Ak. - If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A\-1 (if the [inverse](https://www.cuemath.com/algebra/inverse-of-a-matrix/) of A exists). - If λ is an eigenvalue of A, then \|A\| / λ is an eigenvalue of the [adjoint](https://www.cuemath.com/algebra/adjoint-of-a-matrix/) of A. Apart from these properties, we have another theorem related to eigenvalues called the "**[Caley-Hamilton Theorem](https://www.cuemath.com/algebra/cayley-hamilton-theorem/)"**. It says, "every square matrix satisfies its characteristic equation". i.e., if A is a square matrix then it satisfies \|A - λI\| = 0. For example, if λ2 - 8λ + 12 = 0 is the characteristic equation of a square matrix A, then A2 - 8A + 12 = 0. - Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc. - They are used in the design of car stereo systems. - They are also used to design bridges. - It is not surprising to know that eigenvalues are also used in determining Google's page rank. - They are used in geometric transformations. Read More

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