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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.

thepillsbury
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  • Is there some difference between an orthogonal similarity transformation and a regular similarity transformation? –  Nov 21 '15 at 01:40
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    Here's the usual process for diagonalizing a matrix. –  Nov 21 '15 at 01:41
  • 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 01:49
  • A similarity transformation of $M$ is just $P^{-1}MP$ where $P$ is invertible. An orthogonal one is when $P$ is even an orthogonal matrix. It always exists when the matrix $M$ is real and symmetric, like yours. – Anne Bauval Mar 09 '24 at 10:22
  • The eigenvalues are $1$, with eigenspace the plane $\Pi:x+y+z=0$, and $4$, with eigenspace the line $L=\Pi^\perp$ (directed by the unit vector $e_1:=(1,1,1)/\sqrt3$). Choose any orthonormal basis $(e_2,e_3)$ of $\Pi$, and the matrix $P$ whose columns are $e_1,e_2,e_3$ will do the job. – Anne Bauval Mar 09 '24 at 10:45

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