线性代数常用公式合集

..逆A可r(nA的列（行）向量线性无关A的特征值全不

为0A0Axx，Ax,Ax精品文档放心下载n阵ATA是正定矩AEApppp是初等阵12si○nnn.感谢阅读A不可逆r(nA0A的列（行）向量线性相关0是A的特征值Ax有非零解,其基础解系即为A关于0的特征向量感谢阅读r(aEn○(aEx精品文档放心下载=-a

b向量组等价矩阵等价()有反身性、对称性、传递性矩阵相似()矩阵合同(),e,,e：√e12nnnp教材；,ee12,ee12,,e1；n,,en④trE=n；Dn,ene12,,e.naaa1112aaaaa)2jj21222n()11(anjnjjjjaaa1n2n1jnn2jn212√.s.......谢谢阅读..感谢阅读A与B,则AO=AAOOBOBABBOAA=()AB

ABBOBO.谢谢阅读aa2n1OaaOOaa()a2n11)aa1)aa2nn111x1x21xx222xxn2xxn1jinxxxn1n1n112na11a由mnm行nAija12a222nmn矩阵.记作：A或mna21aam2aaijamnAmn*AAijTA11A12A21A22A2nAAn2，Aij为A.AAnn√:A○：1AAab1db1主换位cdca副变号(AE)(EA1)谢谢阅读初等行变换!未找到引用源。aa12a311a11a2a3a1a2a311a31a21a1.s.......mAnAmn(A)(A)√A精品文档放心下载mnmn,B√设Amnns,A的列向量为,12则ABCms,,,12n,n,,Bbb11bb2122bbn2AxciA,,,12sA,A,,A12sCAB.感谢阅读C的行向量能由B的行向量线性表示，AT.精品文档放心下载a11a21a12a22an2caaa1n11111122a1n2a2n2c1c2cma2n22aaamn2ac1222mnanmaa1m22√○○精品文档放心下载○○.感谢阅读√.谢谢阅读ABTACTT√分块矩阵的转置矩阵：CDBDTTA1A1分块矩阵的逆矩阵：BB1BA1B1AACA11A1BO1AOA11OOBCBBBA*A11,BA22A*B11BABB22AB*B11AAB,AAn111111n22ABn2222A(A*B(BA√(A0(I)B或(II)XAB感谢阅读(I)的解法：构造(AB)初等行变换(EX)谢谢阅读(II)的解法：将等式两边转置化为TXTBT，感谢阅读用(I)的方法求出XXT①.感谢阅读.s.......②.③.谢谢阅读④.感谢阅读⑤p教材.1≤i≤n).⑥中任一向量,,,i12

nn1.⑦,,,12

nn1,,,.i12

n⑧m维列向量组,,,r(n；12

nm维列向量组,,,r(n.12n⑨若,,,12,线性无关，而,,,,则12n,,,12nn.⑩..感谢阅读0谢谢阅读0精品文档放心下载⑪谢谢阅读.感谢阅读.√对A○○乘A；谢谢阅读对A○○乘A.精品文档放心下载Arr1Arr(r感谢阅读向量组,12,,nr(,12,,n)AB.AB精品文档放心下载,,,12n和,12.,,n,,,12n,12,,n⑫A与BB，P,Qr(r(B),BB,感谢阅读.s........A与Br(,,,12n)r(,,,12n)r(,12,n,,,,12n)A与B.⑬,B可由向量组,,,,,1212sn)=,,,)r(,)≤r(,,,r(,,,,,r(,,,).1212ns1212n12sn⑭,且sn,.可由向量组,,,,,,,121212sns则s≤n.,,,线性无关且可由,,,1212sn⑮,且r(,,,)可由向量组,,,,,r(,,,121212s12snn)p教材94,例⑯.感谢阅读⑰.感谢阅读⑱.感谢阅读⑲设A是mn若r(m，A谢谢阅读若r(n，A,,,,12n.√①若AOr(≥1若AOr(00≤r(Amnr(A)r(AT)r(ATA)p教材101,例)≤n)②③r(kA)r(若k0④若A,Bmnns,若r(AB)0r(r(B)nB的列向量全部是Ax0的解⑤r(AB)≤minr(A),r(B)⑥若A可逆r(AB)r(B)若B可逆r(AB)r(.Ax只有零解若r(Amn)nr(AB)r(B)O；ABOBA在矩阵乘法中有左消去律CABACB.s.......r(AB)r(B)若r(B)nB在矩阵乘法中有右消去

律.ns⑧若r(