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Understanding Eigenvalues & Eigenvectors in Matrices

Explore eigenvalues & eigenvectors of matrices, the characteristic equation, finding eigenvectors, and diagonalization concepts. Learn how eigenvalues determine matrix properties.

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Understanding Eigenvalues & Eigenvectors in Matrices

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  1. Eigenvalues & Eigenvectors

  2. 7.1 Eigenvalues & Eigenvectors • If A is an n n matrix, do there exist nonzero vectorx 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

  3. 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 = xThe vector x is called an eigenvector of A corresponding to . • An eigenvector cannot be zero. • An eigenvalue of  = 0 is possible. • Ax = x  (I – A)x = 0This homogeneous system of equations has nonzero solutions iff (I – A) is not invertible, i.e., det(I – A) = 0.

  4. 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. • The equation det(I – A) = 0 is called the characteristic equation of A.

  5. Section 7-1 Characteristic Polynomial • Characteristic polynomial of A:the eigenvalues of an n n matrix A correspond to the roots of the characteristic polynomial of A. • Because the characteristic polynomial of A is of degree n, A can have at most n distinct eigenvalues.

  6. Section 7-1 Finding Eigenvectors • For each eigenvalue i, find the eigenvector corresponding to i by solving the homogeneous system (iI – A)x = 0.This requires row reducing an matrix (iI – 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.

  7. Section 7-1 Example 4 Find the eigenvalues and corresponding eigenvectors of Sol: two eigenvalues: 1,  2

  8. Section 7-1 Example 4 (cont.) • (I – A)x = 0

  9. Section 7-1 Example 4 (cont.) • Method 2: (I – A)x = 0eigenvectors:

  10. 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

  11. 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 , , and . Note that has a multiplicity of 2.

  12. Section 7-1 Theorem 7.3 Eigenvalues for Triangular Matrices If A is an n ntriangular matrix, then its eigenvalues are the entires on its main diagonal. • Its proof follows from the fact that the determinant of a triangular matrix is the product of its diagonal elements.

  13. Section 7-1 Example 7 Find the eigenvalues for the following matrices. (a) (b)

  14. Section 7-1 Linear Transformations • 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 .

  15. Section 7-1 Example 8 Find the eigenvalues and corresponding eigenspaces of Standard basis Sol: the eigenvalues of A are 4, 2, 2.

  16. Section 7-1 Example 8 (cont.) • Basis for eigenspace • Basis for eigenspaces

  17. Section 7-1 Example 8 (cont.) • Let T:R3R3 be the linear transformation whose standard matrix is A, and let be the basis of R3 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.

  18. 7.2 Diagonalization • 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.

  19. Section 7-2 Example 1 The matrix from Example 5 of Section 6.4, is diagonalizable, because has the property that

  20. 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. Hencethey must have the same eigenvalues.

  21. 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:

  22. Section 7-2 Theorem 7.5 Condition for Diagonalizable An n n matrixA is diagonalizable if and only if it has n linearly independent eigenvectors. • Assume that A has n linearly independent eigenvectors p1, p2, …, pn with corresponding eigenvaluesLet

  23. Section 7-2 Example 3 (a) Let . A has the following eigenvalues andcorresponding eigenvectors.

  24. Section 7-2 Example 3 (cont.) (b) Let . A has the following eigenvalues and corresponding eigenvectors.

  25. Section 7-2 Example 4 Show that the following matrix is not diagonalizable. pf: Because A is triangular, the only eigenvalue isEvery eigenvector of A has the formHence A does not have two linearly independent eigenvectors. A is not diagonalizable.

  26. Section 7-2 Example 5 Show that the following matrix is diagonalizable. Then find a matrix P such that is diagonal. pf:Eigenvalues of A are

  27. Section 7-2 Example 5 (cont.) eigenvector

  28. Section 7-2 Example 5 (cont.)

  29. Section 7-2 Theorem 7.6 & Example 7 Sufficient Condition for Diagonalizable If an n n matrixA has ndistincteigenvalues, 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.

  30. Section 7-2 Linear Transformation • 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 R3 such that the matrix for T relative to B is diagonal. (Example 5)

  31. 7.3 Symmetric Matrices and Orthogonal Diagonalization • Def: A square matrix is symmetric if it is equal to its transpose: • Theorem 7.7: Eigenvalues of Symmetric MatricesIf A is an n nsymmetric 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  hask linearly independent eigenvectors. That is, the eigenspace of  has dimension k. • The set of eigenvalues of A is called the spectrum of A.

  32. 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.

  33. Section 7-3 Orthogonal Matrices • Definition: A square matrix P is called orthogonal if it is invertible and • Theorem 7.8: Property of Orthogonal MatricesAn 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.

  34. Section 7-3 Proof of Theorem 7.8 Then the product has the form

  35. 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.

  36. Section 7-3 Example 5 (cont.) Letting produces and Therefore, is an orthonormal set.

  37. Theorem 7.9 Property of Symmetric Matrices Let A be an n nsymmetric matrix. If 1 and 2 are distinct eigenvalues of A, then their corresponding eigenvectors x1 and x2 are orthogonal. pf: Ax1 = 1x1 and Ax2 = 2x2.

  38. Example 6 Show that any two eigenvectors of corresponding to distinct eigenvalues are orthogonal. pf:

  39. Orthogonal Diagonalization • A matrix A is orthogonally diagonalizable if there exists an orthogonal matrixP such that is diagonal. • Theorem 7.10: Fundamental Theorem of Symmetric MatricesLet A be an n n matrix. Then A is orthogonal diagonalizable and has real eigenvalues if and only ifA is symmetric.

  40. Example 8 Find an orthogonal matrix P that orthogonally diagonalizes Sol: 1. Find all eigenvalues. Thus the eigenvalues are and

  41. Example 8 (cont.) 2. Find eigenvector for each eigenvalue of multiplicity 1, and then normalize it. eigenvector

  42. Example 8 (cont.) 3. Construct the orthogonal matrix P. * Verify P is correct by computing

  43. 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

  44. 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 isnotorthonormal, apply the Gram-Schmidt orthonormalization process.Two eigenvectors for 2are and * v1 is orthogonal to v2 and v3.

  45. Example 9 (cont.) Gram-Schimidt process: 4. Construct the orthogonal matrix P.

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