正交矩阵与正定矩阵的关系 谁能给出两个正交 什么是正定矩阵，正交矩阵
\u6b63\u4ea4\u77e9\u9635\u4e0e\u6b63\u5b9a\u77e9\u9635\u7684\u5173\u7cfb\u8bbeM\u662fn\u9636\u5b9e\u7cfb\u6570\u5bf9\u79f0\u77e9\u9635\uff0c \u5982\u679c\u5bf9\u4efb\u4f55\u975e\u96f6\u5411\u91cf
X=(x_1,...x_n) \u90fd\u6709 XMX^t>0\uff0c\u5c31\u79f0M\u6b63\u5b9a\u3002
\u6b63\u5b9a\u77e9\u9635\u5728\u76f8\u4f3c\u53d8\u6362\u4e0b\u53ef\u5316\u4e3a\u6807\u51c6\u578b\uff0c \u5373\u5355\u4f4d\u77e9\u9635\u3002
\u6240\u6709\u7279\u5f81\u503c\u5927\u4e8e\u96f6\u7684\u77e9\u9635\u4e5f\u662f\u6b63\u5b9a\u77e9\u9635\u3002
--------------------------------------
n\u9636\u5b9e\u77e9\u9635 A\u79f0\u4e3a\u6b63\u4ea4\u77e9\u9635\uff0c\u5982\u679c\uff1aA\u00d7A\u2032=I
\u5219\u4e0b\u5217\u8bf8\u6761\u4ef6\u662f\u7b49\u4ef7\u7684:
1) A \u662f\u6b63\u4ea4\u77e9\u9635
2) A\u00d7A\u2032=I \u4e3a\u5355\u4f4d\u77e9\u9635
3) A\u2032\u662f\u6b63\u4ea4\u77e9\u9635
4) A\u7684\u5404\u884c\u662f\u5355\u4f4d\u5411\u91cf\u4e14\u4e24\u4e24\u6b63\u4ea4
5) A\u7684\u5404\u5217\u662f\u5355\u4f4d\u5411\u91cf\u4e14\u4e24\u4e24\u6b63\u4ea4
6) (Ax,Ay)=(x,y) x,y\u2208R
\u5728\u7ebf\u6027\u4ee3\u6570\u91cc\uff0c\u6b63\u5b9a\u77e9\u9635 (positive definite matrix) \u6709\u65f6\u4f1a\u7b80\u79f0\u4e3a\u6b63\u5b9a\u9635\u3002\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u6b63\u5b9a\u77e9\u9635\u7684\u6027\u8d28\u7c7b\u4f3c\u590d\u6570\u4e2d\u7684\u6b63\u5b9e\u6570\u3002\u4e0e\u6b63\u5b9a\u77e9\u9635\u76f8\u5bf9\u5e94\u7684\u7ebf\u6027\u7b97\u5b50\u662f\u5bf9\u79f0\u6b63\u5b9a\u53cc\u7ebf\u6027\u5f62\u5f0f\uff08\u590d\u57df\u4e2d\u5219\u5bf9\u5e94\u57c3\u5c14\u7c73\u7279\u6b63\u5b9a\u53cc\u7ebf\u6027\u5f62\u5f0f\uff09\u3002
\u5982\u679cAAT=E\uff08E\u4e3a\u5355\u4f4d\u77e9\u9635\uff0cAT\u8868\u793a\u201c\u77e9\u9635A\u7684\u8f6c\u7f6e\u77e9\u9635\u201d\uff09\u6216ATA=E\uff0c\u5219n\u9636\u5b9e\u77e9\u9635A\u79f0\u4e3a\u6b63\u4ea4\u77e9\u9635\u3002\u6b63\u4ea4\u77e9\u9635\u662f\u5b9e\u6570\u7279\u6b8a\u5316\u7684\u9149\u77e9\u9635\uff0c\u56e0\u6b64\u603b\u662f\u5c5e\u4e8e\u6b63\u89c4\u77e9\u9635\u3002\u5c3d\u7ba1\u6211\u4eec\u5728\u8fd9\u91cc\u53ea\u8003\u8651\u5b9e\u6570\u77e9\u9635\uff0c\u4f46\u8fd9\u4e2a\u5b9a\u4e49\u53ef\u7528\u4e8e\u5176\u5143\u7d20\u6765\u81ea\u4efb\u4f55\u57df\u7684\u77e9\u9635\u3002
\u6b63\u4ea4\u77e9\u9635\u6bd5\u7adf\u662f\u4ece\u5185\u79ef\u81ea\u7136\u5f15\u51fa\u7684\uff0c\u6240\u4ee5\u5bf9\u4e8e\u590d\u6570\u7684\u77e9\u9635\u8fd9\u5bfc\u81f4\u4e86\u5f52\u4e00\u8981\u6c42\u3002\u6b63\u4ea4\u77e9\u9635\u4e0d\u4e00\u5b9a\u662f\u5b9e\u77e9\u9635\u3002\u5b9e\u6b63\u4ea4\u77e9\u9635\uff08\u5373\u8be5\u6b63\u4ea4\u77e9\u9635\u4e2d\u6240\u6709\u5143\u90fd\u662f\u5b9e\u6570\uff09\u53ef\u4ee5\u770b\u505a\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u9149\u77e9\u9635\uff0c\u4f46\u4e5f\u5b58\u5728\u4e00\u79cd\u590d\u6b63\u4ea4\u77e9\u9635\uff0c\u8fd9\u79cd\u590d\u6b63\u4ea4\u77e9\u9635\u4e0d\u662f\u9149\u77e9\u9635\u3002
\u6269\u5c55\u8d44\u6599\u6b63\u4ea4\u77e9\u9635\u4e0d\u4e00\u5b9a\u662f\u6b63\u5b9a\u77e9\u9635
\u4e3e\u53cd\u4f8b\uff1a
A=-E\uff0c\u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u4f46\u4e0d\u662f\u6b63\u5b9a\u77e9\u9635\u3002
\u6b63\u5b9a\u77e9\u9635\u4e5f\u4e0d\u4e00\u5b9a\u662f\u6b63\u4ea4\u77e9\u9635\u3002
\u77e9\u9635\u6b63\u5b9a\u7684\u524d\u63d0\u662f\u5bf9\u79f0\u9635\uff0c\u800c\u6b63\u4ea4\u77e9\u9635\u4e0d\u4e00\u5b9a\u662f\u5bf9\u79f0\u9635\u3002
\u5c06\u77e9\u9635\u5206\u89e3\u4e3a\u7b80\u5355\u77e9\u9635\u7684\u7ec4\u5408\u53ef\u4ee5\u5728\u7406\u8bba\u548c\u5b9e\u9645\u5e94\u7528\u4e0a\u7b80\u5316\u77e9\u9635\u7684\u8fd0\u7b97\u3002\u5bf9\u4e00\u4e9b\u5e94\u7528\u5e7f\u6cdb\u800c\u5f62\u5f0f\u7279\u6b8a\u7684\u77e9\u9635\uff0c\u4f8b\u5982\u7a00\u758f\u77e9\u9635\u548c\u51c6\u5bf9\u89d2\u77e9\u9635\uff0c\u6709\u7279\u5b9a\u7684\u5feb\u901f\u8fd0\u7b97\u7b97\u6cd5\u3002\u5173\u4e8e\u77e9\u9635\u76f8\u5173\u7406\u8bba\u7684\u53d1\u5c55\u548c\u5e94\u7528\uff0c\u8bf7\u53c2\u8003\u300a\u77e9\u9635\u7406\u8bba\u300b\u3002\u5728\u5929\u4f53\u7269\u7406\u3001\u91cf\u5b50\u529b\u5b66\u7b49\u9886\u57df\uff0c\u4e5f\u4f1a\u51fa\u73b0\u65e0\u7a77\u7ef4\u7684\u77e9\u9635\uff0c\u662f\u77e9\u9635\u7684\u4e00\u79cd\u63a8\u5e7f\u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1-\u6b63\u4ea4\u77e9\u9635
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1-\u6b63\u5b9a\u77e9\u9635
举反例：
A=-E，是正交矩阵，但不是正定矩阵。
正定矩阵也不一定是正交矩阵。
举反例：
A=
-1 0
0 1
，是正交矩阵，但不是正定矩阵。
扩展阅读：为什么正定aii大于0 ... 正交与正定的区别 ... 正定矩阵十大特征 ... 怎么证明半正定 ... 既正交又正定的矩阵 ... 正交化结果唯一吗 ... 正定矩阵一定正交吗 ... 正交矩阵三大特征 ... 判断矩阵正定最简单的办法 ...
