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Hey guys, i'm having trouble trying to understand how the diagonalised matrix is produced

e.g.

A =

1 | 3 | 0

3 | -2 |-1

0 | -1 | 1

I've calculated the eigenvalues to be 1, -4, 3

My question is, how do we know that

D =

1 | 0 | 0

0 | 3 | 0

0 | 0 | -4

and not any other combination of 1, 3, -4 along the diagonal, or does it make no difference?

e.g.

A =

1 | 3 | 0

3 | -2 |-1

0 | -1 | 1

I've calculated the eigenvalues to be 1, -4, 3

My question is, how do we know that

D =

1 | 0 | 0

0 | 3 | 0

0 | 0 | -4

and not any other combination of 1, 3, -4 along the diagonal, or does it make no difference?

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