301_F10_hw7 - AME 301 Homework #7 | Due Fri Oct 22, 2010 1....

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Unformatted text preview: AME 301 Homework #7 | Due Fri Oct 22, 2010 1. A matrix ~ A is called similar to a matrix A if the matrices are related by a similarity transform ~ A = P 1 AP for any nonsingular matrix P . (Here all matrices are n n .) Similar matrices have the same eigenvalues. Demonstrate this property by calculating the eigenvalues of A and ~ A where A = 2 4 1 0 1 0 1 1 1 1 0 3 5 and P = 2 4 1 1 1 3 5 : (The calculation is simpli ed by noting that P is an orthogonal matrix.) 2. Consider the real symmetric matrix A = 2 4 2 1 1 1 2 1 1 1 2 3 5 : (a) Find the eigenvalues and eigenvectors. (For convenience, call the non-repeated eigenvalue 1 and the repeated eigenvalue 2 .) (b) For 2 , verify that the geometric multiplicity is equal to the algebraic multiplicity. (This is guaranteed, since A is real symmetric.) (c) Is the eigenvector s 1 (for 1 ) orthogonal to every possible eigenvector s 2 (for 2 )? Prove your answer by calculating s T 1 s 2 ....
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301_F10_hw7 - AME 301 Homework #7 | Due Fri Oct 22, 2010 1....

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