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 Post subject: diagonal matrixPosted: Mon, 17 Nov 2008 22:19:45 UTC
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Joined: Tue, 7 Oct 2008 19:58:13 UTC
Posts: 30
Hello.

We consider the matrix M= with

I have to find a matrix P orthogonal and symmetric such is diagonal.

I know that M is a symmetric matrix whose entries are real so M can be diagonalized by an orthogonal matrix.

I have shown that the eigenvalues of M are (a+b)² , (a-b)² and a²-b² and is a matrix such

But the problem is that Q is not symmetric ...

Thanks for any help.

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 Post subject: Posted: Tue, 18 Nov 2008 00:04:35 UTC
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Joined: Mon, 23 Jun 2003 17:34:53 UTC
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Location: San Antonio,Texas USA
Have you tried the symmetric part of ?

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 Post subject: Posted: Tue, 18 Nov 2008 21:18:13 UTC
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Joined: Sat, 7 Jan 2006 18:29:24 UTC
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Location: Leeds, UK
The first two columns of Q are determined (at least up to a scalar multiple), because the corresponding eigenvalues have multiplicity 1. Therefore if Q is going to be symmetric the top two rows are also determined, and Q should look like . Now you just have to fill in the *s so that the other two columns are eigenvectors for the repeated eigenvalue a²-b². (Also, you'll then need to divide Q by 2 to make it orthogonal.)

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