# Orthogonal similarity transformation

Can someone please show me how to diagonalize a matrix such as the one below using an orthogonal similarity transformation? $$\begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{bmatrix}$$

I have been looking everywhere online to find an example of orthogonal similarity transformations but I can't find any. Am I searching for the wrong thing? Is there another name for it, because similarity transformations seem awfully close to Jordan canonical form?

Please help. Thank you in advance.

• Is there some difference between an orthogonal similarity transformation and a regular similarity transformation? – user137731 Nov 21 '15 at 1:40
• Here's the usual process for diagonalizing a matrix. – user137731 Nov 21 '15 at 1:41
• I wish I knew the answer to your question..I was hoping someone would explain that to me. Appreciate the feedback. – thepillsbury Nov 21 '15 at 1:49

## 1 Answer

For stuff like that WolframAlpha is a great help: $$\pmatrix{2&1&1\\1&2&1\\1&1&2}=SDS^{-1}= \pmatrix{ -1&-1&1\\ 0&1&1\\ 1&0&1 } \pmatrix{ 1&0&0\\ 0&1&0\\ 0&0&4 } \frac13 \pmatrix{ -1&-1&2\\ -1&2&-1\\ 1&1&1 }$$

• This is not an orthogonal transformation since the inverse of S does not equal the transpose of S. I figured out a way to do it using the Gramm-Schmidt process. – thepillsbury Nov 24 '15 at 21:30
• @thepillsbury yes I know, but I thought you'll figure out the details yourself, whic was correct... – draks ... Nov 26 '15 at 20:29