# The norm of a diagonalizable matrix is its largest eigenvalue?

In relation to the Euclidean norm...

What are the conditions for when this occurs? Is it only real symmetric matrices?

When is this not the case?

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Which norm do you mean? –  Davide Giraudo Sep 19 '12 at 19:43
Euclidean norm... edited not. Thx. –  Dirk Calloway Sep 19 '12 at 19:47
@DirkCalloway Surely you mean the operator norm of the matrix. –  Alex Becker Sep 19 '12 at 19:48
And do you mean the largest in modulus? Because nothing a priori gives that the eigenvalues are non negative, which can of course happen even when the matrix is symmetric. And when you don't assume this, the eigenvalues can be complex and not real. –  Davide Giraudo Sep 19 '12 at 20:05
@DirkCalloway: If the matrix is diagonalizable, you have the largest absolute value of the eigenvalues. Since a positive semidefinite matrix (I assume that's what you meant with SPD) has only positive eigenvalues, for this it is the same as the largest eigenvalue. I don't know what happens for non-diagonalizable matrices, though. –  celtschk Sep 19 '12 at 20:23