Class 31
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Hermitian Matrices and Diagonalization continued. Special case of symmetric
matrics and orthogonal diagonalization.
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Problems:
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Diagonalize the following two matrices using a real **orthogonal** matrix, i.e.
P^TAP=D, D diagonal and real, P real and P^TP=I
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A= [ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]
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B= [ -1 0 1 ]
[ 0 1 0 ]
[ 0 0 1 ]
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Given the **general** n by n matrix A, show that:
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trace(A)=sum of the eigenvalues of A
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det(A)=product of the eigenvalues of A
(Hint: Use diagonalizability or reduction by elementary matrices to a triangular
matrix.)