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Can anybody explain how to orthogonally diagonalize the following matrix:

$$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$

Am I correct in saying the eigenvalues are 10 and -1 and the corresponding eigenvectors are [1,1/sqrt(10)] and [1,-sqrt(10)]

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  • $\begingroup$ yes, wolframalpha.com/input/… and to diagonalize, use a matrix with the eigenvectors and conjugate as $M^{-1}AM$ $\endgroup$ – janmarqz Jan 25 '15 at 15:36
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    $\begingroup$ You can use the process explained in this answer. $\endgroup$ – Git Gud Jan 25 '15 at 15:38
  • $\begingroup$ here is a diagonalisation calculator with steps. $\endgroup$ – Math137 Jan 25 '15 at 15:40
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You can diagonalize as: $$\pmatrix{-1 & 0 \\ 0 & 10}$$

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  • $\begingroup$ Would you be able to explain how to get Q such that (Q^T)AQ where A is the matrix I gave in the question? $\endgroup$ – Gary Jan 25 '15 at 15:40
  • $\begingroup$ @gary see git guds answer. Also note that the diagonal entries are exactly the eigenvalues you already calculated ... $\endgroup$ – Loreno Heer Jan 25 '15 at 15:43

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