hw3sol.pdf - George W Woodruff School of Mechanical...

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George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology ME6401 HW SET #3 ________________________________________________________________________ 1) Let A be an arbitrary n n matrix with eigenvalues i , i=1,…,n and modal matrix M. Prove the following facts: a) The eigenvalues of A T are the same as those of A and the modal matrix A T is M -T . Let J be the Jordan form of A. Since J is diagonal, J T =J thus ܬ ൌ ܯ ିଵ ܣܯ ⇒ ܬ ൌ ܬ ൌ ሺܯ ିଵ ܣܯሻ ൌ ܯ ܣ ܯ ି் . This shows that the eigenvalues of A and A T are the same and the modal matrix of A T is M -T. b) The eigenvalues of A -1 are 1/ i , i=1,…,n, and the modal matrix A -1 is M. ܬ ൌ ܯ ିଵ ܣܯ ⇒ ܬ ିଵ ൌ ሺܯ ିଵ ܣܯሻ ିଵ ൌ ܯܣ ିଵ ܯ ି . This shows that the eigenvalues of A -1 are 1/ i , i=1,…,n and the modal matrix of A -1 is M -1. c) The determinant of A is |A|= ߣ ௜ୀଵ Let J be the Jordan form of A. Since J is diagonal with i ’s on its diagonal, then |J|= ߣ ௜ୀଵ . On the other hand, ܬ ൌ ܯ ିଵ ܣܯ ⇒ ܣ ൌ ܯܬܯ ିଵ ൌ |ܣ| ൌ |ܯ||ܬ||ܯ ିଵ | ൌ |ܯ||ܬ||ܯ| ିଵ ൌ |ܬ| ൌ ∏ ߣ ௜ୀଵ d) The eigenvalues and modal matrix of ܣ ൌ ܶܣܶ ିଵ are i , i=1,…,n, and ܯ ൌ ܶܯ , respectively, for an arbitrary transformation matrix T . ܣ ൌ ܶܣܶ ିଵ ൌ ܶܯܬܯ ିଵ ܶ ିଵ ൌ ሺܶܯሻܬሺܶܯሻ ିଵ . This proves that A and ܣ have the same
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