Chapter Content n n n Eigenvalues and Eigenvectors

Chapter Content n n n Eigenvalues and Eigenvectors Diagonalization Orthogonal Digonalization 2021/2/22 Elementary Linear Algebra 1

7 -3 The Orthogonal Diagonalization Matrix Form n Given an n n matrix A, if there exist an orthogonal matrix P such that the matrix P-1 AP = PTAP then A is said to be orthogonally diagonalizable and P is said to orthogonally diagonalize A. 2021/2/22 Elementary Linear Algebra 2

Theorem 7. 3. 1 & 7. 3. 2 n If A is an n n matrix, then the following are equivalent. q q q n A is orthogonally diagonalizable. A has an orthonormal set of n eigenvectors. A is symmetric. If A is a symmetric matrix, then: q q 2021/2/22 The eigenvalues of A are real numbers. Eigenvectors from different eigenspaces are orthogonal. Elementary Linear Algebra 3

7 -3 Diagonalization of Symmetric Matrices n The following procedure is used for orthogonally diagonalizing a symmetric matrix. q q q 2021/2/22 Step 1. Find a basis for each eigenspace of A. Step 2. Apply the Gram-Schmidt process to each of these bases to obtain an orthonormal basis for each eigenspace. Step 3. Form the matrix P whose columns are the basis vectors constructed in Step 2; this matrix orthogonally diagonalizes A. Elementary Linear Algebra 4

7 -3 Example 1 n Find an orthogonal matrix P that diagonalizes n Solution: q q q 2021/2/22 The characteristic equation of A is The basis of the eigenspace corresponding to = 2 is Applying the Gram-Schmidt process to {u 1, u 2} yields the following orthonormal eigenvectors: Elementary Linear Algebra 5

7 -3 Example 1 q The basis of the eigenspace corresponding to = 8 is Applying the Gram-Schmidt process to {u 3} yields: q Thus, q orthogonally diagonalizes A. 2021/2/22 Elementary Linear Algebra 6