高等代数:设A为实对称阵，证明A可逆的充分必要条件是存在矩阵B，使AB+B^TA为正定矩阵 设A为n阶实对称矩阵，证明：秩（A）=n的充分必要条件为存在...
\u8bbeA\u662fn\u9636\u5b9e\u5bf9\u79f0\u8bc1\u660ea\u53ef\u9006\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5b58\u5728n\u9636\u5b9e\u77e9\u9635b\u4f7f\u5f97AB+B\u8f6c\u7f6eA\u662f\u6b63\u5b9a\u82e5A\u53ef\u9006\uff0c\u53d6B=A1 (A\u7684\u9006\u77e9\u9635) \u5219AB+B`A=2E\uff0c\u547d\u9898\u5f97\u8bc1 \uff08B`\u8868\u793aB\u8f6c\u7f6e\uff09
\u82e5AB+B`A\u6b63\u5b9a\uff0c\u5219\u5bf9\u4e8e\u4efb\u610fX\uff0c0<=X`(AB+B`A)X=X`ABX+X`B`AX=X`A`BX+X`B`AX=(AX\uff0cBX)+(BX,AX)=2(AX,BX) \u82e5AX=0 \u6709\u975e\u96f6\u89e3X0\uff0c\u5c06X0\u5e26\u5165\u4e0a\u5f0f \u5219\u67090<0 \u77db\u76fe \u6240\u4ee5AX=0\u53ea\u6709\u96f6\u89e3 \u6240\u4ee5
r(A)=n A\u53ef\u9006
\u201c\u5fc5\u8981\u6027\u201d\uff08?\uff09\u5229\u7528\u53cd\u8bc1\u6cd5\u8fdb\u884c\u8bc1\u660e\uff0e\u53cd\u8bbe\uff1ar\uff08A\uff09\uff1cn\uff0c\u5219|A|=0\uff0e\u4e8e\u662f\u03bb=0\u662fA\u7684\u7279\u5f81\u503c\uff0c\u5047\u8bbe\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u4e3ax\uff0c\u5373\uff1aAx=0\uff08x\u22600\uff09\uff0c\u6240\u4ee5\uff1axTAT=0\uff0e\u4ece\u800c\uff1axT\uff08AB+BTA\uff09x=xTABx+xTBTAx=0\uff0c\u4e0eAB+BTA\u662f\u6b63\u5b9a\u77e9\u9635\u77db\u76fe\uff0c\u6545\u5047\u8bbe\u4e0d\u6210\u7acb\uff0e\u6240\u4ee5\uff0c\u79e9\uff08A\uff09=n\uff0e\u201c\u5145\u5206\u6027\u201d\uff08?\uff09\u56e0\u4e3a r\uff08A\uff09=n\uff0c\u6240\u4ee5A\u7684\u7279\u5f81\u503c\u03bb1\uff0c\u03bb2\uff0c\u2026\uff0c\u03bbn\u5168\u4e0d\u4e3a0\uff0e\u53d6\u77e9\u9635B=A\uff0c\u5219\uff1aAB+BTA=AA+AA=2A2\uff0c\u5b83\u7684\u7279\u5f81\u503c\u4e3a\uff1a2\u03bb12\uff0c2\u03bb22\uff0c\u2026\uff0c2\u03bbn2\u5168\u90e8\u4e3a\u6b63\uff0c\u6240\u4ee5AB+BTA\u662f\u6b63\u5b9a\u77e9\u9635\uff0e
你好！证明如图，用构造法证明充分性，用反证法证明必要性。经济数学团队帮你解答，请及时采纳。谢谢！
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