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Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix $M = \frac{1}{2}(A + {A^T})$.

Why does $\rho (A) \le {\lambda _{\max }}(M)$?

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    $\begingroup$ This is a special case of Ky Fan's result, which says the real parts of the eignevalues of $A$ are majorized by the eigenvalues of $M$; see [K. Fan, Maximal properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA, 37 (1951), pp. 760–766]. $\endgroup$
    – M. Lin
    Commented Jan 14, 2016 at 2:52

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