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## Homework Statement

Given the matrix A:

4 2 2

2 4 2

2 2 4

Find the matrix P such that P

^{-1}AP is diagonal

## Homework Equations

## The Attempt at a Solution

So I had this question today on a placement exam and it threw me for a loop. I found the eigenvalues to be 2,2, and 8. The eigenvectors are (-1,1,0), (-1, 0, 1), and (1,1,1) respectively. So my understanding was that with any real symmetric matrix it should be possible to write it in this form. However, my understanding was also that it had to have n linear independent eigenvectors which meant no repeated eigenvalues. Which way is it?