3.3 Eigenvalues and Eigenvectors
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1 .. EIGENVALUES AND EIGENVECTORS 27. Eigenvalues and Eigenvectors In this section, we assume A is an n n matrix and x is an n vector... Definitions In general, the product Ax results is another n vector which will have a different magnitude and direction. However, in some cases, a matrix will act on a vector by only changing its magnitude and leaving its direction unchanged or reversed. In other words the new vector is parallel to the original vector that is Ax = λx where λ is a constant. This will not happen for just any λ and x. They come in pairs. For each λ that will work, correspond one or more vectors x. Definition.. A scalar λ is called an eigenvalue or a characteristic value of A if there is a nontrivial solution x of Ax = λx. Such a vector x is called an eigenvector of A corresponding to the eigenvalue λ. Let us note a few things:. Note that the zero vector x is always a solution of Ax = λx, it is why we are looking for nontrivial vectors x. 2. An eigenvalue can be, but not an eigenvector. The geometric meaning of eigenvalues of eigenvectors is the following. If A is the matrix of a linear transformation T : R n R n that is if T (x) = Ax and if λ is an eigenvalue and x the corresponding eigenvector, then the image of x under this transformation is a scalar multiple of x that is it is parallel to x. We will see more interpretations and applications in the exercises section. Definition..2 The eigenspace of A corresponding to the eigenvalue λ is the set of all eigenvectors of A corresponding to λ...2 Computation and Examples We now focus on how to find eigenvalues and eigenvectors. Recall, we are trying to solve Ax = λx. Two things can happen. Ax = λx Ax λx = Ax λix = (A λi) x = Case.. (A λi) is invertible that is det (A λi). The only solution is the trivial solution. This is not of interest to us since we are seeking a nontrivial solution.
2 28 CHAPTER. NUMERICAL LINEAR ALGEBRA Case..4 (A λi) is not invertible that is det (A λi) =. This is the only case in which we can hope to find a solution. Definition..5 det (A λi) is a polynomial of degree n. characteristic polynomial. It is called the Let us note a few things:. Since det (A λi) is a polynomial of degree n, det (A λi) = will have n solutions, real and/or complex. Some of which may be repeated. 2. This means that if A is an n n matrix then it will have n eigenvalues, call them λ, λ 2,..., λ n. Some may be repeated.. If an eigenvalue λ appears only once in the list, it is called simple. 4. If an eigenvalue λ appears k > times in the list, we say that λ has multiplicity k. 5. If λ, λ 2,..., λ k (k n) are the simple eigenvalues in the list, with corresponding eigenvectors x (), x (2),.., x (k), then the eigenvectors are linearly independent. 6. If λ is an eigenvalue with multiplicity k > then λ will have anywhere from to k linearly independent eigenvectors. 7. If x is an eigenvector corresponding to λ then kx is also an eigenvector corresponding to λ. This means that eigenvectors are defined up to a constant. We usually retain the form of the eigenvector which ( is ) the /2 easiest to read or write. For example, instead of keeping we ( ) /2 would keep. MATLAB will return a unit eigenvector, that is an eigenvector of magnitude. 8. More generally, if v, v 2,..., v k are eigenvectors corresponding to the eigenvalue λ then any linear combination of these vectors is also an eigenvector corresponding to λ. To find the eigenvalues and eigenvectors of an n n matrix A, follow these steps:. Find the determinant of A λi that is the characteristic polynomial of A. It will be a polynomial of degree n. 2. Find the roots of the polynomial obtained in step. These will be the eigenvalues.. For each eigenvalue λ found in step 2, solve (A λi) x = to find the corresponding eigenvector(s) x.
3 .. EIGENVALUES AND EIGENVECTORS 29 We illustrate this with a few examples. Example..6 Find the eigenvalues and eigenvectors of A = Eigenvalues: First, we form A λi. ( ) λ A λi = 5 λ Next, we compute det (A λi). det (A λi) = ( λ) (5 λ) ( ) () = λ 2 6λ + 8 ( 5 Next, we find the roots of λ 2 6λ + 8 that is we solve λ 2 6λ + 8 =. This is the same as solving (λ 4) (λ 2) =. So, the roots hence the eigenvalues are λ = 2 and λ = 4. Eigenvectors:. For each value of λ, we solve (A λi) x =, that is we solve (A 2I) x = then (A 4I) x =. Solution of (A 2I) x = : ( ) ( ) ( ) 2 x (A 2I) x = = 5 2 x 2 ( ) ( ) ( ) x = x 2 ( ) x There is not a unique solution. x = will be a solution if x 2 it satisfies x ( = x 2 ). Once( we ) pick an arbitrary value for x 2, say s x 2 = s then = s is an eigenvector. As we noted s earlier eigenvectors( are ) unique up to a constant. Some people will pick the form of s that is the easiest to write. In this case, it is( with) s =. So, the eigenvector corresponding to λ = 2 is x =. Solution of (A 4I) x = : ( ) ( ) ( ) x (A 4I) x = = x 2 ( ) x There is not a unique solution. x = will be a solution if it satisfies x = x 2. Following what we did ( above, ) we see that the eigenvector corresponding to λ = 4 is x = x 2 ).
4 CHAPTER. NUMERICAL LINEAR ALGEBRA Example..7 Find the eigenvalues and eigenvectors of A = Eigenvalues: det (A λi) = 2 λ λ 2 5 λ λ 2 5 λ = (2 λ) = (2 λ) ( λ (5 λ) + 6) = (2 λ) ( λ 2 5λ + 6 ) = (2 λ) (λ 2) (λ ) = (λ 2) 2 (λ ) Thus the eigenvalues are λ = 2 (multiplicity 2) and λ =. Eigenvectors: Case λ = 2: We solve (A 2I) x =. (A 2I) x = 2 2 x x 2 = x s Solving this system gives x = t and x 2 = x = s. So, x = t = t t + s. The eigenvectors associated with λ = 2 are and. Note they are independent. Case λ = : We solve (A I) x =. (A I) x = 2 x x 2 = 2 x We see that x = t, x 2 = 2 x = 2 t and x = x 2 + x = 2 t + t = t t, It follows that x = 2 t = t 2. So, x = 2 is the t eigenvector associated with λ =.
5 .. EIGENVALUES AND EIGENVECTORS.. Eigenvalues and MATLAB The MATLAB function to get the eigenvalues of a matrix is eig. It can be used different ways; we only show a few here. For a complete list, type help eig within MATLAB.. Given an n n matrix A, eig (A) will display the eigenvalues of A. Each eigenvalue will be printed as many times as its multiplicity. 2. Given an n n matrix A, s = eig (A) will find the eigenvalues of A and store them into the n vector s. As above, each eigenvalue will appear as many times as its multiplicity.. Given an n n matrix A, [V D] = eig (A) will find the eigenvalues and eigenvectors of A. The eigenvectors of A will be stored in V as column vectors. So, V is in fact a matrix. The eigenvalues of A will be stored on the diagonal of D, the remaining entries of D being zeros. The eigenvalues will appear in the same order as the eigenvectors. Note that MATLAB will find eigenvectors which are unit vectors (magnitude ). Example..8 If A = 2 2 then:. eig (A) will return 2. This means that is a simple eigenvalue but 2 2 has multiplicity s = eig (A) will set s = [V D] = eig (A) will set V =.482 and D = indicating that the eigenvectors associated with λ = 2 are and.77 and the eigenvector associated with λ = is You will notethese are unit vectors. Another program might have given and as the eigenvectors associated with λ = 2 and 2 as the eigenvector associated with λ =, which is easier to read. Thus
6 2 CHAPTER. NUMERICAL LINEAR ALGEBRA..4 Properties of Eigenvalues and Eigenvectors We list some important properties. algebra book. Their proof can be found in any linear. It will be useful to remember some properties of determinants. (a) det ( A ) = det (A) (b) det (AB) = det (A) det (B) (c) det ( A T ) = det (A) (d) det (ca) = c n det (A) 2. Powers of a matrix: If Ax = λx then A 2 x = A (Ax) = A (λx) = λax = λ 2 x. In general, A n x = λ n x. Eigenvalues of a triangular or diagonal matrix. Remembering that the determinant of a triangular or a diagonal matrix is the product of its entries on the diagonal, we see that the characteristic polynomial of such n a matrix is (a ii λ) hence the eigenvalues are a ii for i =..n. i= 4. Similar Matrices: Recall that two matrices A and B are similar if there exists an invertible matrix P of the same size such that B = P AP. Two similar matrices have the same eigenvalues. 5. Diagonalization. Suppose that v, v 2,..., v n are linearly independent vectors and λ, λ 2,..., λ n are their corresponding eigenvalues. Define P = λ. [v v 2... v n ] (note this is an n n matrix) and D = λ λ n then AP = P D that is A = P DP or D = P AP which means that the eigenvalues of A and D are the same. We can make the following important conclusions: (a) If P is as defined, then P AP is a diagonal matrix. (b) det D = det ( P AP ) = det ( P ) det (A) det (P ) = det (A) hence the determinant of a matrix is the product of its eigenvalues. Similarly, the trace of a matrix is the sum of its eigenvalues. (c) The above implies that a matrix is invertible if and only if none of its eigenvalues is zero. 6. Powers Matrix Revisited: If A = P DP then A 2 = P DP P DP = P D 2 P. Similarly, A n = P D n P.
7 .. EIGENVALUES AND EIGENVECTORS 7. λ is an eigenvalue of A if and only if λ is an eigenvalue of A. 8. If λ = a + bi is an eigenvalue of A with eigenvector v then λ = a bi is also an eigenvalue of A and its corresponding eigenvector is the conjugate of v. 9. Symmetric Matrices always have real eigenvalues...5 Applications of Eigenvalues and Eigenvectors An important application of eigenvalues and eigenvectors is with solving systems of first order differential equations. Google s page ranking algorithm uses a lot of linear algebra, including eigenvalues and eigenvectors. Here is a paper by Bryan and Leise on Google s PageRank algorithm. Eigenvalues for face recognition (eigenfaces). started it all by Turk and Pentland. Here is the paper which..6 Exercises. In the last section of this document, read and understand the paper on Google page ranking. This could be a potential project. 2. In the last section of this document, read and understand the paper on eigenfaces. This could be a potential project.
8 Bibliography [] M. C, S. M, Y. Z, V. C. L, Big data: related technologies, challenges and future prospects, Springer, 24. [2] J. D, Big data, data mining, and machine learning: value creation for business leaders and practitioners, John Wiley & Sons, 24. [] L. G, What s this all about?, Time, 86 (25), pp [4] H. A. K, Big Data: techniques and technologies in geoinformatics, CRC Press, 24. [5] J. N. K, Data-Driven Modeling & Scientific Computation: Methods for Complex Systems and Big Data, Oxford University Press, 2. [6] F. L R. I, Google le nouvel einstein, Science & Vie, 8 (22), pp [7] K.-C. L, H. J, L. T. Y, A. C, Big Data: Algorithms, Analytics, and Applications, CRC Press, 25. [8] T. MP, Will our data drown us, IEEE Spectrum, (25), pp [9] T. P, Giving your body a "check engine" light, IEEE Spectrum, (25), pp [] L. S, Should you get paid for your data, IEEE Spectrum, (25), pp [] E. S, Their prescription: Big data, IEEE Spectrum, (25), pp [2] S. Q. Y, Big data analysis for bioinformatics and biomedical discoveries, CRC Press, 26. 2
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