李永乐线性代数知识结构图

1.代数余子式和余子式的关系：(1)(1)ijijijijijijijijijijijijijijijijijijijijijijijijMAAMMAAMMAAMMAAM++=−=−副对角行列式：副对角元素的乘积(1)2(1)nnnnnnnn−×−；拉普拉斯展开式：AOACAOACAOACAOACABABABABCBOBCBOBCBOBCBOB==、(1)mnmnmnmnCAOACAOACAOACAOAABABABABBOBCBOBCBOBCBOBC==−i范德蒙行列式：大指标减小指标的连乘积；2.AAAA是nnnn阶可逆矩阵：⇔0AAAA≠（是非奇异矩阵）；⇔()rAnrAnrAnrAn=（是满秩矩阵）⇔AAAA的行（列）向量组线性无关；⇔齐次方程组0AxAxAxAx=有非零解；⇔nnnnbRbRbRbR∀∈，AxbAxbAxbAxb=总有唯一解；⇔AAAA与EEEE等价；⇔AAAA可表示成若干个初等矩阵的乘积；⇔AAAA的特征值全不为0；⇔TTTTAAAAAAAA是正定矩阵；⇔AAAA的行（列）向量组是nnnnRRRR的一组基；⇔AAAA是nnnnRRRR中某两组基的过渡矩阵；2.②、111AOAOAOAOAOAOAOAOOBOBOBOBOBOBOBOB−−−⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠；（主对角分块）③、111OAOAOAOAOBOBOBOBBOBOBOBOAOAOAOAO−−−⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠；（副对角分块）④、11111ACACACACAACBAACBAACBAACBOBOBOBOBOBOBOBOB−−−−−⎛⎞−⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠；⑤、11111AOAOAOAOAOAOAOAOCBCBCBCBBCABBCABBCABBCAB−−−−−⎛⎞⎛⎞=⎜⎟⎜⎟−⎝⎠⎝⎠；（拉普拉斯）3.①、0()min(,)mnmnmnmnrAmnrAmnrAmnrAmn×≤≤；②、()()TTTTrArArArArArArArA=；③、若ABABABAB∼，则()()rArBrArBrArBrArB=；④、若PPPP、QQQQ可逆，则()()()()rArPArAQrPAQrArPArAQrPAQrArPArAQrPAQrArPArAQrPAQ===；（可逆矩阵不影响矩阵的秩）⑤、max((),())(,)()()rArBrABrArBrArBrABrArBrArBrABrArBrArBrABrArB≤≤+；（※）⑥、()()()rABrArBrABrArBrABrArBrABrArB+≤+；（※）⑦、()min((),())rABrArBrABrArBrABrArBrABrArB≤；（※）⑧、如果AAAA是mnmnmnmn×矩阵，BBBB是nsnsnsns×矩阵，且0ABABABAB=，则：（※）Ⅰ、BBBB的列向量全部是齐次方程组0AXAXAXAX=解（转置运算后的结论）；Ⅱ、()()rArBnrArBnrArBnrArBn+≤⑨、若AAAA、BBBB均为nnnn阶方阵，则()()()rABrArBnrABrArBnrABrArBnrABrArBn≥+−；4.*()()1()10()1nrAnnrAnnrAnnrAnrArAnrArAnrArAnrArAnrAnrAnrAnrAn=⎧⎪==−⎨⎪−⎩；3.施密特正交化：12(,,,)rrrraaaaaaaaaaaa⋯11babababa=；1222111[,][,]babababababbabbabbabbbbbbbbb=−i121121112211[,][,][,][,][,][,]rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrbabababababababababababababbbbabbbbabbbbabbbbbbbbbbbbbbbbbbbbbbbbbbb−−−−=−−−−ii⋯i;