A6_soln - Math 235 Assignment 6 Solutions 1 Show that the...

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Math 235 Assignment 6 Solutions 1. Show that the following are equivalent for a symmetric matrix A : (1) A is orthogonal (2) A 2 = I (3) All the eigenvalues of A are ± 1 Solution: (1) (2) If A is orthogonal then I = AA T = AA , since A is symmetric. (2) (3) Assume that A 2 = I and λ is an eigenvalue of A with eigenvector ~v . Then A~v = λ~v A 2 ~v = A ( λ~v ) ~v = λA~v ~v = λ 2 ~v Hence λ 2 = 1, so λ = ± 1. (3) (1) If all the eigenvalues of A are ± 1, then since A is symmetric there exists an orthogonal matrix P such that P T AP = D where the diagonal entries of D are the eigen- values of A . Hence since the eigenvalues of A are ± 1 and D is diagonal, we have DD T = I and so I = DD T = ( P T AP )( P T AP ) T = ( P T AP )( P T A T P ) = P T AA T P I = A T A. Thus A is orthogonal. 2. Show that the following are equivalent for two symmetric matrices A and B (1) A and B are similar (2) A and B have the same eigenvalues (3) A and B are orthogonally similar Solution: (1)
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A6_soln - Math 235 Assignment 6 Solutions 1 Show that the...

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