Eigenvalues Eigenvectors 7 1 Eigenvalues Eigenvectors n n
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Eigenvalues & Eigenvectors
7. 1 Eigenvalues & Eigenvectors n n n If A is an n n matrix, do there exist nonzero vector x in Rn such that Ax is a scalar multiple of x? The scalar, denoted by , is called an eigenvalue of the matrix A, and the nonzero vector x is called an eigenvector of A corresponding to . Ax = x x x 2
Section 7 -1 Definition Let A be an n n matrix. The scalar is called an eigenvalue of A if there is a nonzero vector x s. t. Ax = x The vector x is called an eigenvector of A corresponding to . n n n An eigenvector cannot be zero. An eigenvalue of = 0 is possible. Ax = x ( I – A)x = 0 This homogeneous system of equations has nonzero solutions iff ( I – A) is not invertible, i. e. , det( I – A) = 0. 3
Section 7 -1 Theorem 7. 2 Eigenvalues and Eigenvectors of a Matrix Let A be an n n matrix. 1. An eigenvalue of A is a scalar such that det( I – A) = 0. 2. The eigenvectors of A corresponding to are the nonzero solutions of ( I – A)x = 0. n The equation det( I – A) = 0 is called the characteristic equation of A. 4
Section 7 -1 Characteristic Polynomial n Characteristic polynomial of A: the eigenvalues of an n n matrix A correspond to the roots of the characteristic polynomial of A. n Because the characteristic polynomial of A is of degree n, A can have at most n distinct eigenvalues. 5
Section 7 -1 Finding Eigenvectors n n n For each eigenvalue i, find the eigenvector corresponding to i by solving the homogeneous system ( i. I – A)x = 0. This requires row reducing an matrix ( i. I – A). The resulting reduced row-echelon form must have at least one row of zeros. If an eigenvalue 1 occurs as a multiple root (k times) for the characteristic polynomial, then 1 has multiplicity k. This implies that is a factor of the characteristic polynomial and is not a factor of the characteristic polynomial. 6
Section 7 -1 Example 4 Find the eigenvalues and corresponding eigenvectors of Sol: two eigenvalues: 1, 2 7
Section 7 -1 Example 4 (cont. ) n ( I – A)x = 0 8
Section 7 -1 Example 4 (cont. ) n Method 2: ( I – A)x = 0 eigenvectors: 9
Section 7 -1 Example 5 Find the eigenvalues and corresponding eigenvectors for What is the dimension of the eigenspace of each eigenvalue? Sol: eigenvalues = 2, 2, 2 10
Section 7 -1 Example 6 Find the eigenvalues of and find a basis for each of the corresponding eigenspaces. Sol: The characteristic equation of A is Thus the eigenvalues are , has a multiplicity of 2. , and . Note that 11
Section 7 -1 Theorem 7. 3 Eigenvalues for Triangular Matrices If A is an n n triangular matrix, then its eigenvalues are the entires on its main diagonal. n Its proof follows from the fact that the determinant of a triangular matrix is the product of its diagonal elements. 12
Section 7 -1 Example 7 Find the eigenvalues for the following matrices. (a) (b) 13
Section 7 -1 Linear Transformations n n A number is called an eigenvalue of a linear transformations T: V V if there is a nonzero vector x such that T(x) = x. The vector x is called an eigenvector of T corresponding to , and the set of all eigenvectors of (with the zero vector) is called the eigenspace of . 14
Section 7 -1 Example 8 Find the eigenvalues and corresponding eigenspaces of Standard basis Sol: the eigenvalues of A are 4, 2. 15
Section 7 -1 Example 8 (cont. ) n Basis for eigenspaces 16
Section 7 -1 Example 8 (cont. ) n n Let T: R 3 be the linear transformation whose standard matrix is A, and let be the basis of R 3 made up of the three linearly independent eigenvectors found in Example 8. Then , the matrix of T relative to the basis , is diagonal. Nonstandard basis The main diagonal entires of are the eigenvalues of A. 17
7. 2 Diagonalization n n Diagonalization problem: For a square matrix A, does there exist an invertible matrix P such that is diagonal? Two square matrices A and B are called similar if there exists an invertible matrix P such that Matrices that are similar to diagonal matrices are called diagonalizable. Definition: An n n matrix A is diagonalizable if A is similar to a diagonal matrix. That is, A is diagonalizable if there exists an invertible matrix P such that is a diagonal matrix. 18
Section 7 -2 Example 1 The matrix from Example 5 of Section 6. 4, is diagonalizable, because has the property that 19
Section 7 -2 Theorem 7. 4 Similar Matrices Have the Same Eigenvalues If A and B are similar n n matrices, then they have the same eigenvalues. pf: A and B have the same characteristic polynomial. Hence they must have the same eigenvalues. 20
Section 7 -2 Example 2 The following matrices are similar. and Find the eigenvalues of A and D. Sol: Eigenvalues of D are Because A and D are similar, A has the same eigenvalues. CHECK: 21
Section 7 -2 Theorem 7. 5 Condition for Diagonalizable An n n matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. n Assume that A has n linearly independent eigenvectors p 1, p 2, …, pn with corresponding eigenvalues Let 22
Section 7 -2 Example 3 (a) Let . A has the following eigenvalues and corresponding eigenvectors. 23
Section 7 -2 Example 3 (cont. ) (b) Let . A has the following eigenvalues and corresponding eigenvectors. 24
Section 7 -2 Example 4 Show that the following matrix is not diagonalizable. pf: Because A is triangular, the only eigenvalue is Every eigenvector of A has the form Hence A does not have two linearly independent eigenvectors. A is not diagonalizable. 25
Section 7 -2 Example 5 Show that the following matrix is diagonalizable. Then find a matrix P such that pf: is diagonal. Eigenvalues of A are 26
Section 7 -2 Example 5 (cont. ) eigenvector 27
Section 7 -2 Example 5 (cont. ) 28
Section 7 -2 Theorem 7. 6 & Example 7 Sufficient Condition for Diagonalizable If an n n matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linear independent and A is diagonalizable. Example 7: Determine whether the following matrix is diagonalizable. From Thm 7. 6, A is diagonalizable. 29
Section 7 -2 Linear Transformation n n For a linear transformation T: V V, does there exist a basis B for V such that the matrix T relative to B is diagonal? The answer is “yes, ” provided that the standard matrix for T is diagonalizable. Example 8: Find a basis for R 3 such that the matrix for T relative to B is diagonal. (Example 5) 30
7. 3 Symmetric Matrices and Orthogonal Diagonalization n Def: A square matrix is symmetric if it is equal to its transpose: Theorem 7. 7: Eigenvalues of Symmetric Matrices If A is an n n symmetric matrix, then the following properties are true: 1. A is diagonalizable. 2. All eigenvalues of A are real. 3. If is an eigenvalue of A with multiplicity k, then has k linearly independent eigenvectors. That is, the eigenspace of has dimension k. The set of eigenvalues of A is called the spectrum of A. 31
Section 7 -3 Example 3 Find the eigenvalues of symmetric matrix Determine the dimensions of corresponding the eigenspaces. Sol: The eigenvalues of A are and Because each of these eigenvalues has a multiplicity of 2, the corresponding eigenspace also have dimension 2. 32
Section 7 -3 Orthogonal Matrices n n Definition: A square matrix P is called orthogonal if it is invertible and Theorem 7. 8: Property of Orthogonal Matrices An n n matrix P is orthogonal if and only if its column vectors form an orthonormal set. pf: Suppose that the column vectors of P form an orthonormal set. 33
Section 7 -3 Proof of Theorem 7. 8 Then the product has the form 34
Section 7 -3 Example 5 Show that is orthogonal by showing that Then show that the column vectors of P form an orthonormal set. pf: Therefore P is orthogonal. 35
Section 7 -3 Example 5 (cont. ) Letting produces and Therefore, is an orthonormal set. 36
Theorem 7. 9 Property of Symmetric Matrices Let A be an n n symmetric matrix. If 1 and 2 are distinct eigenvalues of A, then their corresponding eigenvectors x 1 and x 2 are orthogonal. pf: Ax 1 = 1 x 1 and Ax 2 = 2 x 2. 37
Example 6 Show that any two eigenvectors of corresponding to distinct eigenvalues are orthogonal. pf: 38
Orthogonal Diagonalization n n A matrix A is orthogonally diagonalizable if there exists an orthogonal matrix P such that is diagonal. Theorem 7. 10: Fundamental Theorem of Symmetric Matrices Let A be an n n matrix. Then A is orthogonal diagonalizable and has real eigenvalues if and only if A is symmetric. 39
Example 8 Find an orthogonal matrix P that orthogonally diagonalizes Sol: 1. Find all eigenvalues. Thus the eigenvalues are and 40
Example 8 (cont. ) 2. Find eigenvector for each eigenvalue of multiplicity 1, and then normalize it. eigenvector 41
Example 8 (cont. ) 3. Construct the orthogonal matrix P. * Verify P is correct by computing 42
Example 9 Find an orthogonal matrix P that orthogonally diagonalizes Sol: 1. Find all eigenvalues. Thus the eigenvalues are and 1 has a multiplicity of 1 and 2 has a multiplicity of 2 43
Example 9 (cont. ) 2. Find eigenvector for eigenvalue of multiplicity 1, and then normalize it. An eigenvector for 1 is which normalizes to 3. Find eigenvector for eigenvalue of multiplicity k 2. If this set is not orthonormal, apply the Gram-Schmidt orthonormalization process. Two eigenvectors for 2 are and * v 1 is orthogonal to v 2 and v 3. 44
Example 9 (cont. ) Gram-Schimidt process: 4. Construct the orthogonal matrix P. 45
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