Chapter Content n n n Eigenvalues and Eigenvectors

Chapter Content n n n Eigenvalues and Eigenvectors Diagonalization Orthogonal Digonalization 2021/10/29 Elementary Linear Algebra 1

7 -2 Diagonalization n A square matrix A is called diagonalizable q q n if there is an invertible matrix P such that P-1 AP is a diagonal matrix (i. e. , P-1 AP = D); the matrix P is said to diagonalize A. Theorem 7. 2. 1 q If A is an n n matrix, then the following are equivalent. n A is diagonalizable. n A has n linearly independent eigenvectors. 2021/10/29 Elementary Linear Algebra 2

7 -2 Procedure for Diagonalizing a Matrix n The preceding theorem guarantees that an n n matrix A with n linearly independent eigenvectors is diagonalizable, and the proof provides the following method for diagonalizing A. q Step 1. Find n linear independent eigenvectors of A, say, p 1, p 2, …, pn. q q Step 2. From the matrix P having p 1, p 2, …, pn as its column vectors. Step 3. The matrix P-1 AP will then be diagonal with 1, 2, …, n as its successive diagonal entries, where i is the eigenvalue corresponding to pi, for i = 1, 2, …, n. 2021/10/29 Elementary Linear Algebra 3

7 -2 Example 1 n n Find a matrix P that diagonalizes Solution: q From the previous example, we have the following bases for the eigenspaces: n = 2: q Thus, q Also, 2021/10/29 = 1: Elementary Linear Algebra 4

7 -2 Example 2 (A Non-Diagonalizable Matrix) n Find a matrix P that diagonalizes n Solution: q The characteristic polynomial of A is q The bases for the eigenspaces are n q = 1: = 2: Since there are only two basis vectors in total, A is not diagonalizable. 2021/10/29 Elementary Linear Algebra 5

7 -2 Theorems n Theorem 7. 2. 2 q n If v 1, v 2, …, vk, are eigenvectors of A corresponding to distinct eigenvalues 1, 2, …, k, n then {v 1, v 2, …, vk} is a linearly independent set. Theorem 7. 2. 3 q If an n n matrix A has n distinct eigenvalues n then A is diagonalizable. 2021/10/29 Elementary Linear Algebra 6

7 -2 Example 3 n Since the matrix n has three distinct eigenvalues, Therefore, A is diagonalizable. Further, n for some invertible matrix P, and the matrix P can be found using the procedure for diagonalizing a matrix. 2021/10/29 Elementary Linear Algebra 7

7 -2 Example 4 (A Diagonalizable Matrix) n Since the eigenvalues of a triangular matrix are the entries on its main diagonal (Theorem 7. 1. 1). n Thus, a triangular matrix with distinct entries on the main diagonal is diagonalizable. n For example, is a diagonalizable matrix. 2021/10/29 Elementary Linear Algebra 8

7 -2 Example 5 (Repeated Eigenvalues and Diagonalizability) n Whether the following matrices are diagonalizable? 2021/10/29 Elementary Linear Algebra 9

7 -2 Geometric and Algebraic Multiplicity n If 0 is an eigenvalue of an n n matrix A q q then the dimension of the eigenspace corresponding to 0 is called the geometric multiplicity of 0, and the number of times that – 0 appears as a factor in the characteristic polynomial of A is called the algebraic multiplicity of A. 2021/10/29 Elementary Linear Algebra 10

Theorem 7. 2. 4 (Geometric and Algebraic Multiplicity) n If A is a square matrix, then : q q For every eigenvalue of A the geometric multiplicity is less than or equal to the algebraic multiplicity. A is diagonalizable if and only if the geometric multiplicity is equal to the algebraic multiplicity for every eigenvalue. 2021/10/29 Elementary Linear Algebra 11

7 -2 Computing Powers of a Matrix n n If A is an n n matrix and P is an invertible matrix, then (P-1 AP)k = P 1 Ak. P for any positive integer k. If A is diagonalizable, and P-1 AP = D is a diagonal matrix, then P-1 Ak. P = (P-1 AP)k = Dk Thus, Ak = PDk. P-1 The matrix Dk is easy to compute; for example, if 2021/10/29 Elementary Linear Algebra 12

7 -2 Example 6 (Power of a Matrix) n Find A 13 2021/10/29 Elementary Linear Algebra 13

Exercise Set 7. 2 Question 2 14

Exercise Set 7. 2 Question 15 15

Exercise Set 7. 2 Question 20 16

Exercise Set 7. 2 Question 25 17