Class 30
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Hermitian Matrices and Diagonalization continued.
(Sections 6.4)
Beware: The book treats only the special case that A is real and
symmetric.
We are treating the more general case of complex matrices that are Hermitian.
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Theorems:
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All the roots of A are real.
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A is always diagonalizable,----- P^{-1}AP=D (D diagonal and real)
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A can always be diagonalized using a unitary matrix.,
----- P^{-1}AP=D (D diagonal and real, \bar{P}^TP=I)
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Problems:
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Diagonalize the following matrix using a real **orthogonal** matrix, i.e.
P^TAP=D, D diagonal and real, P real and P^TP=I
- (from the text)
A= [ 0 2 2 ]
[ 2 0 2 ]
[ 2 2 0 ]
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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.)