||.||为酉不变范数,对于酉矩阵U和V满足||UAV||=||A||,证明||A||>||A|| 酉矩阵何时能表示成XY（X
\u77e9\u9635\u8303\u6570\u7684\u9149\u4e0d\u53d8\u8303\u6570\u5b9a\u4e49\uff1a\u5982\u679c\u8303\u6570\u2551\u00b7\u2551\u6ee1\u8db3\u2551A\u2551=\u2551UAV\u2551\u5bf9\u4efb\u4f55\u77e9\u9635A\u4ee5\u53ca\u9149\u77e9\u9635U,V\u6210\u7acb\uff0c\u90a3\u4e48\u8fd9\u4e2a\u8303\u6570\u79f0\u4e3a\u9149\u4e0d\u53d8\u8303\u6570\u3002\u3000\u5bb9\u6613\u9a8c\u8bc1\uff0c2-\u8303\u6570\u548cF-\u8303\u6570\u662f\u9149\u4e0d\u53d8\u8303\u6570\u3002\u56e0\u4e3a\u9149\u53d8\u6362\u4e0d\u6539\u53d8\u77e9\u9635\u7684\u5947\u5f02\u503c\uff0c\u6240\u4ee5\u7531\u5947\u5f02\u503c\u5f97\u5230\u7684\u8303\u6570\u662f\u9149\u4e0d\u53d8\u7684\uff0c\u6bd4\u59822-\u8303\u6570\u662f\u6700\u5927\u5947\u5f02\u503c\uff0cF-\u8303\u6570\u662f\u6240\u6709\u5947\u5f02\u503c\u7ec4\u6210\u7684\u5411\u91cf\u76842-\u8303\u6570\u3002\u3000\u53cd\u8fc7\u6765\u53ef\u4ee5\u8bc1\u660e\uff0c\u6240\u6709\u7684\u9149\u4e0d\u53d8\u8303\u6570\u90fd\u548c\u5947\u5f02\u503c\u6709\u5bc6\u5207\u8054\u7cfb\uff1a\u3000\u5b9a\u7406(Von Neumann\u5b9a\u7406)\uff1a\u5728\u9149\u4e0d\u53d8\u8303\u6570\u548c\u5bf9\u79f0\u5ea6\u89c4\u51fd\u6570(symmetric gauge function)\u4e4b\u95f4\u5b58\u5728\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u3000\u4e5f\u5c31\u662f\u8bf4\u4efb\u4f55\u9149\u4e0d\u53d8\u8303\u6570\u4e8b\u5b9e\u4e0a\u5c31\u662f\u6240\u6709\u5947\u5f02\u503c\u7684\u4e00\u4e2a\u5bf9\u79f0\u5ea6\u89c4\u51fd\u6570\u3002
\u9149\u77e9\u9635\u5c31\u662f\u6b63\u4ea4\u77e9\u9635\u5728\u590d\u6570\u57df\u4e0a\u7684\u63a8\u5e7f
由||.||可以定义出一个向量范数c(.): c(x)=|| [x,x,...,x] ||, 那么c(.)是一个酉不变向量范数对于任何向量x, 存在酉阵H使得Hx=||x||_2e_1, 所以c(x)=c(Hx)=||x||_2*c(e_1), 即c(.)和向量2-范数只相差一个常数倍.
然后用c(.)可以定义出一个诱导范数N(.): N(A)=sup c(Ax)/c(x)=sup ||Ax||_2/||x||_2, 即N(.)事实上就是矩阵2-范数
最后
||A||_2 = N(A)
= sup c(Ax)/c(x)
= sup || A[x,x,...,x] || / || [x,x,...,x] ||
<= sup ||A||*|| [x,x,...,x] ||/|| [x,x,...,x] ||
= ||A||
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