## Eigenvalues of sum of commuting matrices [closed]

With reference to the following thread :

http://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums

## Answer by Jonas Meyer is as follows :

If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of the summands. This would be true more generally for commuting normal matrices. For arbitrary positive matrices, the largest eigenvalue of the sum will be less than or equal to the sum of the largest eigenvalues of the summands.

Can you suggest a reference or source for this.

I am trying to prove that the eigen values of a matrix are all negative and if the above statement holds then it will be of great use. So, please point the references for the same.

Thanks in advance, Ramya

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The point is that commuting normal matrices can be simultaneously diagonalized. This is known as the Spectral Theorem, and you can find it on Wikipedia or any reasonable undergraduate linear algebra text. – MTS Jun 26 2010 at 7:44
Closed as 'answer is on wikipedia', per MTS' comment. – Scott Morrison Jun 26 2010 at 18:27