# Orthogonally diagonalizing a matrix

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)]

• yes, wolframalpha.com/input/… and to diagonalize, use a matrix with the eigenvectors and conjugate as $M^{-1}AM$ Jan 25, 2015 at 15:36
• You can use the process explained in this answer. Jan 25, 2015 at 15:38
• here is a diagonalisation calculator with steps. Jan 25, 2015 at 15:40

## 1 Answer

You can diagonalize as: $$\pmatrix{-1 & 0 \\ 0 & 10}$$

• 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?
– Gary
Jan 25, 2015 at 15:40
• @gary see git guds answer. Also note that the diagonal entries are exactly the eigenvalues you already calculated ... Jan 25, 2015 at 15:43