设A为n阶矩阵，且
\u8bbeA\u4e3an\u9636\u77e9\u9635\u5fc5\u8981\u6027:
r(A)=1, \u5219A\u6709\u4e00\u4e2a\u975e\u96f6\u884c,\u4e0d\u59a8\u8bbe\u4e3a\u7b2c\u4e00\u884c, \u4e14\u5176\u4f59\u884c\u662f\u7b2c1\u884c\u7684\u500d\u6570
\u8bb0 A \u7684\u7b2c\u4e00\u884c\u4e3a (b1,b2,...bn)
\u5176\u4f59\u884c\u5206\u522b\u662f a1=1\u500d, a2\u500d,...,an\u500d
\u5219 A = \u03b1^T\u03b2
\u5145\u5206\u6027:
\u4e00\u65b9\u9762 A\u22600,\u6709r(A)>=1
\u53e6\u4e00\u65b9\u9762 r(A)=r(\u03b1^T\u03b2) <= r(\u03b1) = 1
\u6240\u4ee5 r(A)=1.
k\u4e3a\u5e38\u6570\u65f6\uff0c/kA/=k^n*/A/
\u56e0\u6b64\u4e0a\u5f0f=/A/^n*/A^T/\uff0c
\u53c8\u56e0\u4e3a/A^t/=/A/
\u56e0\u6b64 =/A/^\uff08n+1\uff09
所以x=t(1,1,...,1)^T是一族解
又因为r(A)=N-1
所以Ax=0的解是一维子空间（x=t(1,1,...,1)^T就是一维的）
所以通解为x=t(1,1,...,1)^T
绛旓細杩欎笉鏄鏄剧劧鐨勫悧锛屽鏋A涓嶆弧绉╁垯A涓嶅彲閫嗭紝涓嶢^2=E鐭涚浘
绛旓細杩欓噷鐢ㄥ埌浜嗙煩闃电殑绉╃殑涓嶇瓑寮忕殑涓や釜鎬ц川1銆乺(A+B)<=r(A)+r(B)2銆佽嫢n闃剁煩闃礎,B婊¤冻AB=0,鍒 r(A)+r(B)<=n 鎵浠ワ紝鏈棶棰樹腑锛屽洜涓 A^2=A,鎵浠 A(A-E)=0 r(A)+r(A-E)<=n 鍙坮(A)+r(A-E)=r(A)+r(E-A)>= r(A+E-A)=r(E)=n 鎵浠 r(A)+r(A-E)=E ...
绛旓細detA=0,鎴杁etA=1.鐢辨瘡涓厓绱犵瓑浜庡畠鍦╠etA涓殑浠ｆ暟浣欏瓙寮,鍒橝绛変簬瀹冪殑浼撮殢鐭╅樀A*,鍗矨*=A,鐢盇A*=detA*E,鍏朵腑E鏄鍗曚綅闃.鏁卍et(AA*)=detA*detE,det(AA)=detA*detE,detAdetA=detA,detA(detA-1)=0,鏁 detA=0,鎴杁etA=1.
绛旓細鍥犱负A鏄瀹炲绉闃碉紝瀹冧竴瀹氱浉浼间簬瀵硅闃点傛墍浠鐨勭З绛変簬鍏堕潪闆剁壒寰佸肩殑涓暟銆傝岀敱浜嶢^2=2A锛岀壒寰佸嘉婚兘婊¤冻位^2=2位锛屾墍浠ノ诲彧鑳芥槸2鍜0銆侫鐨勭壒寰佸兼湁r涓2鍜n-r涓0锛屾墍浠鐨勮抗锛堢壒寰佸间箣鍜岋級涓2r銆
绛旓細瑙: 鍥犱负 A^2=A, 鎵浠 A(A-E)=0 鎵浠 A 鐨勭壒寰佸煎彧鑳芥槸 0, 1 鍙堝洜涓A鏄痭闃瀹炲绉鐭╅樀, r(A) = r 鎵浠 A 鐨勭壒寰佸兼湁r涓1, n-r涓0 鎵浠 2E-A 鐨勭壒寰佸兼湁r涓1, n-r涓2 鎵浠 |2E-A| = 2^(n-r)
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绛旓細棣栧厛纭畾AX=0鐨勫熀纭瑙ｇ郴鎵鍚悜閲忕殑涓暟.鍥犱负 R(A)=N-1 鎵浠 AX=0鐨勫熀纭瑙ｇ郴鎵鍚悜閲忕殑涓暟涓 N-r(A) = N-(N-1) = 1.鍙堝洜涓篈鐨勫悇琛屽厓绱犱箣鍜屽潎涓洪浂, 鎵浠 (1,1,...,1)' 鏄疉X=0鐨勮В.鎵浠 (1,1,...,1)' 鏄疉X=0鐨勫熀纭瑙ｇ郴.鏁 AX=0 鐨勯氳В涓 k(1,1,...,1)',...
绛旓細棣栧厛, 鍚岃В鍒欏熀纭瑙ｇ郴鎵鍚悜閲忕殑涓暟鐩稿悓 鏁 n-r(A) = n-r(A^3)鎵浠 R(A)=R(A^3)鍏舵, 鍥犱负 R(A)=R(A^3)鎵浠 涓や釜鏂圭▼缁勭殑鍩虹瑙ｇ郴鎵鍚悜閲忕殑涓暟鐩稿悓 鑰 AX=0 鐨勮В 鏄 A^3X=0 鐨勮В (鏄剧劧)鎵浠 AX=0 鐨勫熀纭瑙ｇ郴 涔熸槸 A^3X=0 鐨勫熀纭瑙ｇ郴 鎵浠ヤ袱涓柟绋嬬粍鍚岃В ...
