5. (Friedland) 给定 $A\in M_n$, $\lm_i\in \bbC$, $i=1,\cdots,n$. 证明: 存在对角矩阵 $D\in M_n$ 使得 $\sigma(A+D)=\sed{\lm_1,\cdots,\lm_n}$, 并且满足上述条件的对角矩阵 $D$ 只有有限多个.

证明: 见 [S. Friedland, Matrices with prescribed off-diagonal elements, Israel J. Math., 11 (1972), 184--189].

[詹兴致矩阵论习题参考解答]习题5.5