Find orthogonal matrix $P$ such that $P^tAP = D$ for $A$ is a normal operator

Question

For

$$A = \;\;\; \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \end{pmatrix} $$

there is a real orthogonal matrix $P$ such that $P^tAP = D$ is diagonal. Find such a diagonal matrix $D$.

I know as $A$ is a normal operator, there is a real diagonal matrix corresponding to $A$. Firstly, I tried to find eigenvalue and eigen vectors to find orthogonal matrix $P$, however, such a $P$ is not orthogonal.

Can anybody help? Thanks!

Since you are only asked about the matrix $D$ (and not $P$), you only need the eigenvalues. — Vedran Šego, Dec 08 '13 at 13:09
Is the diagonal matrix I have to find a matrix having diagonal elements as eigenvalues? — Block Jeong, Dec 08 '13 at 13:14
Do you want to find an orthogonal matrix $P$ and a diagonal matrix $D$, or is $D$ sufficient? — Git Gud, Dec 08 '13 at 13:14
Dear Git Gud, $D$ is sufficient as the problem asked :) , and I wonder $D = P^tAP$ is same as diagonal matrix made by chaging basis. — Block Jeong, Dec 08 '13 at 13:17
Yes, the diagonal elements of $D$ are the eigenvalues (in any order you want, although descending is quite common). — Vedran Šego, Dec 08 '13 at 13:18

Amzoti · Answer 1 · 2013-12-08T14:10:56.097

4

Hints:

Find the eigenvalues
Find the eigenvectors
Normalize the eigenvectors and take $P$ to be these normalized eigenvectors, which gives $P^{-1} = P^t$

This is possible since this is a real, symmetric matrix that has three distinct eigenvalues. If that were not the case, this algorithm would need to be modified slightly and this process is not always possible.

For this problem, we have three real and distinct eigenvalues, so we can straight off write the diagonal matrix as:

$$J = \begin{bmatrix} 0 & 0 & 0 \\ 0 & \dfrac{1}{2}\left(9-\sqrt{105}\right) & 0 \\ 0 & 0 & \dfrac{1}{2}\left(9+\sqrt{105}\right) \\ \end{bmatrix} $$

edited Dec 08 '13 at 14:10

answered Dec 08 '13 at 13:15

Amzoti

56,093

The OP said that he doesn't need $P$. – Vedran Šego Dec 08 '13 at 13:19
1

Well done! (Or rare, however you like your steak!) +1 – amWhy Dec 09 '13 at 00:00

Find orthogonal matrix $P$ such that $P^tAP = D$ for $A$ is a normal operator

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