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I've been working on this problem, and I think that I almost have the solution, but I'm not quite there.

Suppose that $A \in M_n(\mathbb C)$ has $n$ distinct eigenvalues $\lambda_1... \lambda_n$. Show that $$\sqrt{\sum _{j=1}^{n} \left | {\lambda_j} \right |^2 } \leq \left \| A \right \|_F\,.$$

I tried using the Schur decomposition of $A$ and got that $\left \| A \right \|_F = \sqrt{TT^*}$, where $A=QTQ^*$ with $Q$ unitary and $T$ triangular, but I'm not sure how to relate this back to eigenvalues and where the inequality comes from.

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1 Answer 1

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You are in the right way. The corresponding Schur decomposition is $A = Q U Q^*$, where $Q$ is unitary and $U$ is an upper triangular matrix, whose diagonal corresponds to the set of eigenvalues of $A$ (because $A$ and $U$ are similar). Now because Frobenius norm is invariant under unitary matrix multiplication:

$$||QA||_F = \sqrt{\text{tr}((QA)^*(QA))} = \sqrt{\text{tr}(A^*Q^* QA)} = \sqrt{\text{tr}(A^*A)} = ||A||_F$$

(the same remains for multiplication of $Q$ on the right) then we could write: $$||A||_F = ||Q U Q^*||_F = ||U||_F \rightarrow \sqrt{\sum_{j=1}^n |\lambda_j|^2} \leq ||A||_F$$

directly proves your statement.

Note: The inequality comes from the definition of the Frobenius norm: The sum of the square of all entries in the matrix. Since $U$ contains the eigenvalues on his diagonal, the term in the left has to be less or equal to the sum over all entries, because $U$ could have non zero entries over his diagonal.

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  • $\begingroup$ Could you expound a bit on your last step, how you got the inequality at the end? I don't understand how that follows. $\endgroup$
    – 100001
    Commented Apr 18, 2016 at 2:40
  • $\begingroup$ @MathLogic clarified in the note. $\endgroup$
    – mavillan
    Commented Apr 18, 2016 at 2:52
  • $\begingroup$ Took me a while to get what's going on here, but I think it is with equality if $A$ is symmetric, but in general if we instead use the Jordan form (rather than the Schur), then it can have nontrivial offdiagonal values that contribute to the frobenius norm. $\endgroup$
    – Y. S.
    Commented Jun 4, 2019 at 19:55

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