[问题2014S15]  设 OO<script id="MathJax-Element-1" type="math/tex">O</script> 为 nn<script id="MathJax-Element-2" type="math/tex">n</script> 阶正交阵，A=\mathrm{diag}\{a_1,a_2,\cdots,a_n\}A=diag{a1,a2,?,an}<script id="MathJax-Element-3" type="math/tex">A=\mathrm{diag}\{a_1,a_2,\cdots,a_n\}</script> 为实对角阵, 证明: 方阵 OAOA<script id="MathJax-Element-4" type="math/tex">OA</script> 的特征值 \lambda_jλj<script id="MathJax-Element-5" type="math/tex">\lambda_j</script> 适合不等式:  m\leq |\lambda_j|\leq M,\,\,1\leq j\leq n, 

m≤|λj|≤M,1≤j≤n,

m=min1≤i≤n|ai|,M=max1≤i≤n|ai|.