Then is $S = \{ A \in M_n (C) : A = A^*$ and $\rho (A) ≤ 1\}$ is compact in $M_n (\mathbb{R})$?
Let $A \in M_n (\mathbb{C})$ and let $\rho (A) = max\{ \mid {\Lambda} : \Lambda$ is an eigenvalue of A \} $
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general-topology
compactness
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0is the last $M_n(\Bbb R)$ supposed to be $M_n(\Bbb C)$? – 2017-01-19
1 Answers
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Hint: by the Spectral Theorem, the spectral radius is equal to the (operator) norm for Hermitian matrices.