# Orthogonal similarity transformation of a 3x3 matrix

my problem is I am not sure how to do an orthogonal similarity transform. I have been told that one form of an orthogonal similarity transform is doing a rotation, but I am not familiar with that. This is the matrix I want to transform.

$$\begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{bmatrix}$$

Would it be acceptable to use the equation D=P^-1 AP? Where D is a diagonal matrix from the eigenvalues of the original matrix A and P^-1 is equal to transpose of P. In other words, can I find matrix P by finding the eigenvalues (finding D) and setting up a system of equations using D=P^TAP? Would this count as an orthogonal similarity transformation?

EDIT------------------------------

So what I currently have is I found the eigenvalues and the eigenvectors to fulfill A=PDP^-1 where D were the eigenvalues and P was made of the eigenvectors and I found the eigenvectors using gramschmit method. Then I normalized the vectors. Would this make it an orthogonal similarity transformation?

• Is your matrix $P$ orthogonal? – amd Nov 22 '15 at 8:17
• Yes, it should be I think. – thepillsbury Nov 22 '15 at 16:12
• Check that it is, and you’re done. – amd Nov 22 '15 at 19:53