Proof problem(Linear Algebra- Eigenvalues/Eigenvectors)

  • #1

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?
 

Answers and Replies

  • #2
Dick
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Sound right to me.
 
  • #3
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
Dick
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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?
 
  • #6
Dick
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oh, thanks!
Well, it is true that "orthogonally diagonalizable" iff symmetric.
 

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