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**[SOLVED] diagonalization, eigenvectors, eigenvalues**

## Homework Statement

Find a nonsingular matrix P such that (P^-1)*A*P is diagonal

| 1 2 3 |

| 0 1 0 |

| 2 1 2 |

## Homework Equations

I am at a loss on how to do this. I've tried finding the eigen values but its getting me nowhere

## The Attempt at a Solution

i row reduced to 1 2 3

0 1 0

0 0 -4

and found : (x-1)^2 and (x+4), which gives me eigen values of 1 and -4.

Using A - Ix, when x = -4, the matrix becomes 5 2 3 0

0 5 0 0

2 1 6 0

Using row reduction: 1 2/5 1/5 0

0 5 0 0

2 1 6 0

1 2/5 1/5 0

0 1 0 0

0 1/5 28/5 0

1 2/5 1/5 0

0 1 0 0

0 0 28/5 0

1 0 0 0

0 1 0 0

0 0 1 0

which means X = y = z = 0, but that's wrong

When i use x = 1: 0 2 3 0

0 0 0 0

2 1 1 0

this gives 2y = -3z

2X + y + z = 0 ==> 2X = z/2

If z = 4, then x = 1, y = -6

That's one correct answer, but i can't get the second one, which is suppose to be

X = -3, y = 0, z = 2