已知A是实反对称矩阵,证明I-A^2为正定矩阵
\u8bbeA,B\u4e3a\u6b63\u5b9a\u77e9\u9635,\u8bc1\u660eA+B\u4e3a\u6b63\u5b9a\u77e9\u9635.\u77e9\u9635A\u662f\u6b63\u5b9a\u7684 \u7b49\u4ef7\u4e8e \u5bf9\u4e8e\u4efb\u610f\u975e\u96f6\u5411\u91cfa,\u90fd\u6709a'Aa>0;
\u5982\u679cA\u3001B\u90fd\u662f\u6b63\u5b9a\u7684\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4efb\u610f\u975e\u96f6\u5411\u91cfa,\u90fd\u6709a'Aa>0;a'Ba>0;
\u663e\u7136\u5bf9\u4e8e\u4efb\u610f\u975e\u96f6\u5411\u91cfa,\u5c31\u6709a'(A+B)a>0;
\u6240\u4ee5A+B\u4e5f\u662f\u6b63\u5b9a\u7684\uff01
\u53ea\u8981\u4f60\u641e\u6e05\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb\u5c31\u884c\u4e86\uff0c\u6700\u597d\u7528\u53cd\u6b63\u6cd5\u8bc1\u4e00\u4e0b\u3002
\u5728\u5b9e\u6570\u8303\u56f4\u5185\uff1a
A\u4e3an\u9636\u7684\u6b63\u5b9a\u77e9\u9635\uff0c\u5219A\u7684n\u4e2a\u7279\u5f81\u503c\u5747\u4e3a\u6b63\u6570 \u7b49\u4ef7\u4e8e \u5bf9\u4e8e\u4efb\u4e00n\u7ef4\u5217\u5411\u91cfx\uff0c\u90fd\u6709x[T]Ax>0,x[T]\u8868\u793aA\u7684\u8f6c\u7f6e\u3002
\u56e0\u6b64\u6709\uff0cx[T]Ax>0\uff0cx[T]Bx>0\uff0c\u76f8\u52a0\u5f97\uff1ax[T](A+B)x>0
\u5373\u5f97A+B\u4e5f\u4e3a\u6b63\u5b9a\u77e9\u9635\u3002
\u5728\u590d\u6570\u8303\u56f4\u5185\uff1a
A\u4e3an\u9636\u7684\u6b63\u5b9a\u77e9\u9635\uff0c\u5219A\u7684n\u4e2a\u7279\u5f81\u503c\u5747\u4e3a\u6b63\u6570 \u7b49\u4ef7\u4e8e \u5bf9\u4e8e\u4efb\u4e00n\u7ef4\u5217\u5411\u91cfx\uff0c\u90fd\u6709x[H]Ax>0,x[H]\u8868\u793aA\u7684\u5171\u8f6d\u8f6c\u7f6e\uff08\u79f0\u4e3aA\u7684Hemite\u77e9\u9635\uff09\u3002
\u56e0\u6b64\u6709\uff0cx[H]Ax>0\uff0cx[H]Bx>0\uff0c\u76f8\u52a0\u5f97\uff1ax[H](A+B)x>0
\u5373\u5f97A+B\u4e5f\u4e3a\u6b63\u5b9a\u77e9\u9635\u3002
\u5148\u7ea6\u5b9a\u7b26\u53f7\u77e9\u9635A\u7684\u8f6c\u7f6e\u8bb0\u4e3aAT
\u5df2\u77e5\uff1aA\u662f\u53cd\u5bf9\u79f0\u9635\uff0c\u5373AT=-A
\u60f3\u8981\u8bc1\u660e\u77e9\u9635E-A^2\u4e3a\u6b63\u5b9a\u9635\uff0c\u9996\u5148\u8981\u8bf4\u660e\u5b83\u662f\u5bf9\u79f0\u9635\uff1a
\u77e9\u9635E-A^2=E+A\u00d7(-A)=E+A\u00d7AT,\u8fd9\u662f\u4e00\u4e2a\u5bf9\u79f0\u9635\uff0c\u53ef\u4ee5\u8bc1\u660eE+A\u00d7AT\u7684\u8f6c\u7f6e\u5c31\u662f\u5b83\u672c\u8eab
\u56e0\u4e3a\uff08E+A\u00d7AT\uff09T=ET+(A\u00d7AT)T=E+(AT)T \u00d7 AT=E+A\u00d7AT
\u5176\u6b21\u518d\u6765\u8bc1\u660eE-A^2=E+A\u00d7AT\u4e3a\u6b63\u5b9a\u9635\uff0c\u8fd9\u9700\u8981\u7528\u5230\u4ee5\u4e0b\u4e00\u4e2a\u7ed3\u8bba\uff1a
\u4e24\u4e2a\u6b63\u5b9a\u77e9\u9635\u7684\u548c\u4ecd\u7136\u662f\u6b63\u5b9a\u9635\uff08\u76f4\u63a5\u5229\u7528\u4e8c\u6b21\u578b\u7ae0\u8282\u4e2d\u6b63\u5b9a\u9635\u7684\u5b9a\u4e49\uff0c\u5c31\u53ef\u4ee5\u8bc1\u660e\uff09
\u5229\u7528\u8fd9\u4e2a\u7ed3\u8bba\uff0c\u5982\u679c\u8bc1\u660e\u4e86E\u548cA\u00d7AT\u90fd\u662f\u6b63\u5b9a\u9635\uff0c\u90a3\u4e48\u5b83\u4eec\u7684\u548cE+A\u00d7AT\uff0c\u5373E-A^2\u5c31\u662f\u6b63\u5b9a\u9635
\u6613\u5f97\uff0cE\u7684\u6b63\u5b9a\u6027
\u4e0b\u9762\u8bc1\u660eA\u00d7AT\u662f\u6b63\u5b9a\u9635\uff0c\u4f9d\u636e\u6b63\u5b9a\u7684\u5b9a\u4e49
\u5bf9\u4e8e\u4efb\u610f\u7684\u975e\u96f6\u5411\u91cfy\uff0cyT\u00d7\uff08A\u00d7AT\uff09\u00d7y=\uff08yT\u00d7A\uff09\u00d7\uff08AT\u00d7y\uff09=\uff08AT\u00d7y\uff09T \u00d7(AT\u00d7y)\u8fd9\u662f\u5411\u91cf(AT\u00d7y)\u548c\u6b64\u5411\u91cf\u7684\u8f6c\u7f6e\uff08AT\u00d7y\uff09T\u4e4b\u4e58\u79ef\uff0c\u662f\u5b9e\u6570\u7684\u5e73\u65b9\u548c\uff0c\u4e00\u5b9a\u5927\u4e8e\u96f6\uff0c
\u56e0\u6b64A\u00d7AT\u662f\u6b63\u5b9a\u9635
所以 I-A^2 的特征值为 1 或 1-(ki)^2 = 1+k^2 >0
所以 I-A^2 是正定矩阵
绛旓細璇佹槑锛氳A涓哄疄鍙嶅绉扮煩闃碉紝位鏄畠鐨勪换鎰忎竴涓壒寰佹牴锛岃 鏄睘浜庣壒寰佹牴位鐨勪竴涓壒寰佸悜閲忥紝鍗 涓鏂归潰锛屾湁 鍙︽柟闈紝鍙堟湁 鏁 浣嗘槸 鏁 鍗澄讳负闆舵垨绾櫄鏁般
绛旓細鍙嶄緥: A=-I 娉ㄦ剰, D^2鐨勭壒寰佸间竴瀹氶潪璐, 鑰屾潯浠跺苟娌℃湁璇碅鍗婃瀹 鐢氳嚦瀵逛簬鍗婃瀹氶樀A=0, 涔熸病鏈瀵圭О涓嶅畾鐨勫钩鏂规牴
绛旓細鍙杧1=1,xj=0,j鈮1锛屽垯f(x)=a11=0.鍚岀悊鍙杋=2,3,鈥︹,n寰楀埌a22=a33=鈥︹=ann=0 鍙堝彇xi=xj=1(i鈮爅),鍏朵粬涓洪浂锛屽垎鍒护i,j鍙栭亶1鍒皀鐨勪笉鍚屽硷紝f(x)=aij+aji=0,鎵浠ij=-aji锛宨鈮爅 浜庢槸aij=-aij瀵逛换鎰1<=i,j<=n閮芥垚绔嬶紝鍗A鏄弽瀵圭О鐭╅樀 鍙嶄箣锛岃嫢A^T=-A,鍒檉(x)=f...
绛旓細鍙嶅绉扮煩闃婊¤冻aij=-aji,鍏朵富瀵硅涓婂厓绱鏄痠=j鐨勶紝鎵浠ユ湁aii=-aii锛鏁卆ii=0锛宨=1,2,...,n
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绛旓細婊¤冻A^T=-A鐨瀹炵煩闃A灏卞彨瀹炲弽瀵圭О闃点傛瘮濡 0 1 2 -1 0 -3 -2 3 0 鍏冪礌aij閮芥槸瀹炴暟锛屽苟涓攁ij=-aji(i锛j=1锛2锛屸)锛宯鐨刵闃鐭╅樀A=(aij)銆傚畠鏈変互涓嬫ц川锛1.A鐨勭壒寰佸兼槸闆舵垨绾櫄鏁帮紱2.|A|鏄竴涓潪璐熷疄鏁扮殑骞虫柟锛3.A鐨勭З鏄伓鏁帮紝濂囨暟闃鍙嶅绉扮煩闃鐨勮鍒楀紡绛変簬闆 銆
绛旓細璁A涓n闃瀹炲弽瀵圭О鐭╅樀锛r涓篈鐨勭壒寰佸硷紝x涓篈瀵瑰簲r鐨勭壒寰佸垪鍚戦噺 A*x=r*x (x鐨勫叡杞浆缃煩闃)*A*x=r*(x鐨勫叡杞浆缃煩闃)*x鈥︹︹憼 鍥犱负x闈為浂锛屾墍浠(x鐨勫叡杞浆缃煩闃)*x鏄竴涓鏁帮紝璁颁负X 灏嗏憼寮忎袱杈瑰垎鍒綔鍏辫江杞疆锛屽洜涓篈瀹炲弽瀵圭О锛屾墍浠鐨勫叡杞浆缃煩闃=-A (x鐨勫叡杞浆缃煩闃)*(-A)*x...
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绛旓細鍓嶆彁, B鏄疄鐨鍙嶅绉闃 濡傛灉I+B涓嶅彲閫, 閭ｄ箞瀛樺湪闈為浂鍚戦噺x浣垮緱(I+B)x=0, 杩欐牱x^T(I+B)x=0 浣嗘槸x^T(I+B)x=x^Tx>0, 鐭涚浘
绛旓細鍙嶇О鐨勭煩闃典篃灏鏄弽瀵圭О鐭╅樀銆傚弽瀵圭О鐭╅樀鏄寚A= - AT(A鐨勮浆缃墠鍔犺礋鍙) 瀹冪殑绗厾琛屽拰绗厾鍒楀悇鏁扮粷瀵瑰肩浉绛夛紝绗﹀彿鐩稿弽銆 浜庢槸锛屽浜庡瑙掔嚎鍏冪礌锛孉(i,i)=-A(i锛宨),鏈2A(i锛宨)=0锛 鍦ㄩ潪鍋舵暟鍩熶腑锛屾湁A(i锛宨)=0銆傚弽瀵圭О鐭╅樀瀹氫箟鏄:A= - A鈥(A鐨勮浆缃墠鍔犺礋鍙)锛屽畠鐨勭i琛屽拰绗琲...
