怎么证明反对称矩阵的特征值只能是令或纯虚数？！ 实反对称矩阵的特征值只能为零或纯虚数怎么证
\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u201c\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f\u96f6\u6216\u865a\u6570\u201d\u5982\u4f55\u8bc1\u660e\u5462?\u8bc1\u660e\uff1a\u8bbeA\u4e3a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u03bb\u662f\u5b83\u7684\u4efb\u610f\u4e00\u4e2a\u7279\u5f81\u6839\uff0c\u800c
\u662f\u5c5e\u4e8e\u7279\u5f81\u6839\u03bb\u7684\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\uff0c\u5373
\u4e00\u65b9\u9762\uff0c\u6709
\u53e6\u65b9\u9762\uff0c\u53c8\u6709
\u6545
\u4f46\u662f
\u6545
\u5373\u03bb\u4e3a\u96f6\u6216\u7eaf\u865a\u6570\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\u6027\u8d281\uff1a\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u503c\u4e3a0\u3002
\u6027\u8d282\uff1a\u5f53A\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u65f6\uff0cXTAX =0\u3002
\u6027\u8d283\uff1a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f\u96f6\u6216\u7eaf\u865a\u6570\u3002
\u6027\u8d284\uff1a\u82e5A\u4e3a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cA\u7684\u7279\u5f81\u503c\u03bb= bi(b\u22600)\u6240\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u03b1+\u03b2i\u4e2d\u5b9e\u90e8\u4e0e\u865a\u90e8\u5bf9\u5e94\u7684\u5411\u91cf\u03b1\u3001\u03b2\u76f8\u4e92\u6b63\u4ea4
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635
\u8bbeA\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cr\u4e3aA\u7684\u7279\u5f81\u503c\uff0cx\u4e3aA\u5bf9\u5e94r\u7684\u7279\u5f81\u5217\u5411\u91cf
A*x=r*x
(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*A*x=r*(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*x\u2026\u2026\u2460
\u56e0\u4e3ax\u975e\u96f6\uff0c\u6240\u4ee5(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*x\u662f\u4e00\u4e2a\u6b63\u6570\uff0c\u8bb0\u4e3aX
\u5c06\u2460\u5f0f\u4e24\u8fb9\u5206\u522b\u4f5c\u5171\u8f6d\u8f6c\u7f6e\uff0c\u56e0\u4e3aA\u5b9e\u53cd\u5bf9\u79f0\uff0c\u6240\u4ee5A\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635=-A
(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*(-A)*x=(r\u7684\u5171\u8f6d)*X
-(x\u7684\u5171\u8f6d\u8f6c\u7f6e\u77e9\u9635)*A*x=(r\u7684\u5171\u8f6d)*X\u2026\u2026\u2461
\u5c06\u2460\u2461\u4e24\u5f0f\u76f8\u52a0\uff0c (r+r\u7684\u5171\u8f6d)*X=0
\u56e0\u4e3aX>0\uff0c\u6240\u4ee5r+r\u7684\u5171\u8f6d=0
\u5373r=0\u6216r\u662f\u7eaf\u865a\u6570
\u6269\u5c55\u8d44\u6599\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\u6027\u8d281\uff1a\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u503c\u4e3a0\u3002
\u6027\u8d282\uff1a\u5f53A\u4e3an\u9636\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u65f6\uff0c \u6709XTAX =0\u3002
\u6027\u8d283\uff1a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u662f\u96f6\u6216\u7eaf\u865a\u6570\u3002
\u6027\u8d284\uff1a\u82e5A\u4e3a\u5b9e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cA\u7684\u7279\u5f81\u503c\u03bb= bi(b\u22600)\u6240\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u03b1+\u03b2i\u4e2d\u5b9e\u90e8\u4e0e\u865a\u90e8\u5bf9\u5e94\u7684\u5411\u91cf\u03b1\u3001\u03b2\u76f8\u4e92\u6b63\u4ea4\u3002
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对任一向量都有x'Ax=0(因为x'Ax是一个数,数的转置是它本身,就有x'Ax=(x'Ax)'=x'A'x=-x'Ax,看等式两边),尤其x为特征向量时也成立,则ax'x=x'Ax=0.其中x为非零向量.
同理A的共轭也是反对称阵,且特征值为a共轭,对应特征向量为x共轭,就有a共轭x'共轭x共轭=0
由ax'x=0,则a为0,或纯虚数(这要考虑x为复向量时,x'x的情况才能得出结论).
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绛旓細璁続鍙嶇О锛屼笖AX=位X,(X!=0)鍒欙紙X鐨勫叡杞浆缃)AX=位锛圶鐨勫叡杞浆缃)X=位|X|^2 涓よ竟鍙栬浆缃紝骞舵敞鎰忓埌A瀹炲弽绉帮紝鍒欐湁 -锛圶鐨勫叡杞浆缃)AX=位锛圶鐨勫叡杞浆缃)X=锛埼荤殑鍏辫江锛墊X|^2 涓ゅ紡鐩稿姞寰楋細銆愇+锛埼荤殑鍏辫江锛夈*|X|^2=0 鍥犱负X鏄鐗瑰緛鍚戦噺锛岋紒=0锛屾墍浠ワ細銆愇+锛埼荤殑鍏辫江锛夈=0 ...
绛旓細璁続锛孊涓哄弽瀵圭О鐭╅樀锛孉B涓嶄竴瀹氭槸鍙嶅绉扮煩闃点璁続涓哄弽瀵圭О鐭╅樀锛岃嫢A鐨勯樁鏁颁负濂囨暟锛屽垯A鐨勮鍒楀紡涓0锛汚鐨勯樁鏁颁负鍋舵暟锛屽垯鏍规嵁鍏蜂綋鎯呭喌璁＄畻銆
绛旓細杩欎釜涓嶉毦.鍙嶅绉扮煩闃A,婊¤冻A'=-A,璁綼涓篈鐨勭壒寰佸,x涓哄搴旂壒寰佸悜閲.鍒欐槸Ax=ax.瀵逛换涓鍚戦噺閮芥湁x'Ax=0(鍥犱负x'Ax鏄竴涓暟,鏁扮殑杞疆鏄畠鏈韩,灏辨湁x'Ax=(x'Ax)'=x'A'x=-x'Ax,鐪嬬瓑寮忎袱杈),灏ゅ叾x涓虹壒寰佸悜閲忔椂涔熸垚绔,鍒檃x'x=x'Ax=0.鍏朵腑x涓洪潪闆跺悜閲.鍚岀悊A鐨勫叡杞篃鏄弽瀵圭О闃,涓...
绛旓細璁続涓簄闃跺疄鍙嶅绉扮煩闃锛宺涓篈鐨勭壒寰佸锛寈涓篈瀵瑰簲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...
绛旓細涓嶅Θ璁炬瀹鍙嶅绉扮煩闃涓篈,鍏跺睘浜鐗瑰緛鍊位鐨勭壒寰鍚戦噺涓篨,鍗矨X=位X锛庝袱绔乏涔榅H,鍙緱XHAx=位XHX锛庝袱绔啀鍙栧叡杞浆缃,骞跺埄鐢ˋ涓哄疄鍙嶅绉扮煩闃,鍙緱锛峏HAX=位XHX锛庝粠鑰屾湁(位锛嵨)XHX=0锛庡洜涓篨鈮0,鎵浠HX鈮0,浜庢槸鏈壩伙紞位=0,鍗澄讳负闆舵垨绾櫄鏁帮紟
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绛旓細鍙浼璇佹槑Hermite鐭╅樀鐨勭壒寰佸閮芥槸瀹炴暟灏辫浜嗐傚鏋淗鏄疕ermite鐭╅樀锛(c,x)鏄疕鐨勭壒寰佸锛屽嵆Hx=cx锛岄偅涔坈=x*Hx/(x*x)鏄疄鏁般傛帴涓嬫潵锛孉鏄弽Hermite鐭╅樀褰撲笖浠呭綋iA鏄疕ermite鐭╅樀锛屾墍浠ュ弽Hermite鐭╅樀鐨勭壒寰佸奸兘鍦ㄨ櫄杞翠笂锛屽疄鍙嶅绉扮煩闃褰撶劧鏄弽Hermite鐭╅樀銆傚綋鐒朵篃鍙互鐩存帴瀵笰x=cx杩涜澶勭悊寰楀埌conj(c...
绛旓細Proof:Suppose A is a reel skew-symmetric matrix,and 位 is a eigenvalue of A.That is, A伪=位伪 锛埼=(a1,a2,...,an)'锛墂e multply by 锛埼卞叡杞級鈥檕n both sides (伪鍏辫江)'A伪=(伪鍏辫江)'位伪=位(伪鍏辫江)'伪 on the other hand (伪鍏辫江)'A伪=(伪鍏辫江)'(-A')伪=-(A...
绛旓細A鏄疄鍙嶅绉扮煩闃 => A鏄弽Hermite鐭╅樀 <=> iA鏄疕ermite鐭╅樀(i鏄櫄鏁板崟浣)娉ㄦ剰Hermite闃电殑鐗瑰緛鍊閮芥槸瀹炴暟, 鎵浠鐨勭壒寰佸煎彧鑳鍦ㄨ櫄杞翠笂 绗簩棰樻槸绗竴棰樼殑鏄剧劧鎺ㄨ 鑷充簬绗笁棰, 鍙互鐢℉ermite闃电殑璋卞垎瑙ｅ姞涓婄涓棰樼殑缁撹鏉ュ仛, 涔熷彲浠ョ洿鎺ョ敤涔樻硶楠岃瘉QQ^T=I ...
