# Proof problem(Linear Algebra- Eigenvalues/Eigenvectors)

1. Dec 11, 2011

### foxofdesert

1. The problem statement, all variables and given/known data
True/False
The geometric multiplicity of an eigenvalue of a symmetric matrix necessarily equals to its algebric multiplicity.

2. Relevant equations

3. The attempt at a solution
True.
If a matrix is symmetric, then the matrix is diagonalizable. Since the matrix is diagonalizable, there must be eigenvectors correspond to each eigenvalues.

So, I did the proof, but I'm not so sure if it sounds right. I think there could be something more tricky or missing. Would you guys check if this sounds right to you?

2. Dec 11, 2011

### Dick

Sound right to me.

3. Dec 11, 2011

### foxofdesert

Thanks for checking. Just quick checking tho,
'A matrix is symmetric if and only if the matrix is diagonalizable.'

Is this a right statement?

or 'orthogonally diagonalizable'

4. Dec 11, 2011

### Dick

No, it's not iff. Is the matrix [[1,1],[0,0]] diagonalizable? Is it symmetric?

5. Dec 11, 2011

### foxofdesert

oh, thanks!

6. Dec 11, 2011

### Dick

Well, it is true that "orthogonally diagonalizable" iff symmetric.