diagonal matrix

Zac · **Joined:** Tue, 7 Oct 2008 19:58:13 UTC **Posts:** 30

Hello.

We consider the matrix M= $\left( \begin{array}{cccc} a^2 & ab & ab & b^2\\ ab & a^2 & b^2 & ab \\ ab & b^2 & a^2 & ab \\ b^2 & ab & ab & a^2 \\ \end{array} \right)$ with $a,b \in \mathbb{R}$

I have to find a matrix P orthogonal and symmetric such $P^{-1} M P$ 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 $Q= \left( \begin{array}{cccc} 1 & 1 & 0 & 1\\ 1 & -1 & 1 & 0 \\ 1 & -1 & -1 & 0 \\ 1 & 1 & 0 & -1 \\ \end{array} \right)$ is a matrix such
$Q^{-1} M Q=\left( \begin{array}{cccc} (a+b)^2 & 0 & 0 & 0\\ 0 & (a-b)^2 & 0 & 0 \\ 0 & 0 & a^2-b^2 & 0 \\ 0 & 0 & 0 & a^2-b^2 \\ \end{array} \right)$
But the problem is that Q is not symmetric ...

Thanks for any help. :roll:

royhaas · **Posted:** Tue, 18 Nov 2008 00:04:35 UTC

Have you tried the symmetric part of

?

Opalg · **Posted:** Tue, 18 Nov 2008 21:18:13 UTC

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 $\begin{bmatrix}1&1&1&1\\ 1&-1&-1&1\\ 1&-1&*&*\\ 1&1&*&*\end{bmatrix}$ . 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.)

S.O.S. Mathematics CyberBoard

diagonal matrix

Who is online