If a is an eigenvalue of A. show that a^(-1) is an eigenvalue of A^-1. I don't know where to start.
Last edited by Jhevon; Mar 30th 2009 at 09:13 PM. Reason: Restored problem
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Originally Posted by lord12 If x is an eigenvalue of A. show that a^(-1) is an eigenvalue of A^-1. I don't know where to start. Do you mean that if is an eigenvalue of , show that is an eigenvalue of ?
yes!
Originally Posted by lord12 yes! If is an eigenvalue of , then for some eigenvector . Notice that , where is the Identity matrix. Therefore . For a nonzero . Therefore . If is nonsingular, then . By definition of inverses . Therefore is an eigenvalue of .
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