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For the matrix $$\begin{pmatrix} 2 & 2 \\ 2 & 2 \\ \end{pmatrix}$$

Find a matrix that orthogonally diagonalizes A. Find a matrix such that $P^t$$AP$ is diagonal.

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  • $\begingroup$ The columns of $P$ are the normalised eigenvectors of $A$. $\endgroup$ – user1551 Mar 28 '17 at 18:36
  • $\begingroup$ If you’d gotten the correct eigenvectors of this matrix in your previous question, you’d be done. $\endgroup$ – amd Mar 28 '17 at 21:03
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The eigenvectors are easily seen to be $(1,1)$ and $(1,-1)$, which gives $$ P = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, $$ and $$ P^{T} \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} P = \begin{pmatrix} 4 & 0 \\ 0 & 0 \end{pmatrix}. $$

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Find the eigenvectors and construct a matrix P with colums of P the eigenvectors. This P will give you what you want.

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  • $\begingroup$ No, this won't always work. We need $P$ to be orthogonal. Just putting the eigenvectors together is not enough. $\endgroup$ – user1551 Mar 28 '17 at 18:42

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