# 606 - *Proposition: If A = AT Rnn, then all eigenvalues of...

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1 * Proposition: If A = A T R n × n , then all eigenvalues of A are real. Proof Let A x = λ x , where x = [ x 1 x 2 " x n ] T 0 . Since a ij = a ji , 1 11 1 1 0, () nn i ii i i ij i j i j ii j n TT ij i j i i ij i a xx a xx xx axx A xx λλ == = = ≠∈ ++ ∑∑ === xx x x ±²³ ²´ ±²²³²²´ R R * Proposition: If A = A H C n × n , then all eigenvalues of A are real. Proof Let A x = λ x , where x = [ x 1 x 2 " x n ] T 0 . Since 1 1 1 , and , i ii i i ij i j ij j i j ii ii ij ji n ij i j i i i axx axx aa A = = =∈ = x R R R Example: Consider 22 2 2 2 det( ) ( ) . Since ( ) 4( ) ( ) 4 0, has two real eigenvalues. T ab AA bc tI A t a c t ac b ac a cb b A × ⎡⎤ ⎢⎥ ⎣⎦ ⇒− = + + +− −= −+ R Proof Let u , v R n be eigenvectors of A corresponding to eigenvalues λ , µ , respectively. ( λ−µ ) u v = 0 u v = 0 since λ−µ≠ 0. 14 any A = A T R n × n * Theorem 6.14' If u and v are eigenvectors of any A = A H C n × n that correspond to distinct eigenvalues, then u and v are orthogonal. Proof Follow the proof for Theorem 6.14 , with R changed to C and A T changed to A H . 15 real matrix A = A T

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2 Proof Sufficiency (“if”): A = ( P T ) 1 DP 1 = ( P 1 ) 1 DP T = PDP T A T = ( PDP T ) T = ( P T ) T DP T = PDP T = A . Necessity * (“only if”): By induction on n . The necessity obviously holds for n = 1. Assume the necessity holds for n 1, and consider A R ( n +1) × ( n +1) . A has an eigenvector b 1 R n +1 corresponding to a real eigenvalue λ .
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## This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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606 - *Proposition: If A = AT Rnn, then all eigenvalues of...

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