Proof problem(Linear Algebra- Eigenvalues/Eigenvectors)

[foxofdesert]

Homework Statement

True/False
The geometric multiplicity of an eigenvalue of a symmetric matrix necessarily equals to its algebric multiplicity.

Homework Equations

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?

 

[Dick]

Sound right to me.

 

[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'

 

[Dick]

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'

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

 

[foxofdesert]

oh, thanks!

 

[Dick]

oh, thanks!

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