线性代数复习笔记

∣ A ∣ = ∑ j = 1 n a i j A i j ( i = 1 , 2 , . . . , n ) = ∑ i = 1 n a i j A i j ( j = 1 , 2 , . . . , n ) ; A i j = ( − 1 ) i + j M i j . |A|=\sum_{j=1}^na_{ij}A_{ij}\ (i=1,2,...,n)=\sum_{i=1}^na_{ij}A_{ij}\ (j=1,2,...,n);\ A_ij=(-1)^{i+j}M_{ij}. ∣A∣=∑j=1naijAij (i=1,2,...,n)=∑i=1naijAij (j=1,2,...,n); Aij=(−1)i+jMij.
∑ j = 1 n a i j A k j = ∑ i = 1 n a i j A i k = 0 , i ≠ k . \sum_{j=1}^na_{ij}A_{kj}=\sum_{i=1}^na_{ij}A_{ik}=0,\ i\ne k. ∑j=1naijAkj=∑i=1naijAik=0, i=k.

副对角线: ∣ B n ∣ = ( − 1 ) n ( n − 1 ) 2 ∏ i = 1 n b i , n + 1 − i |B_n|=(-1)^\frac{n(n-1)}{2}\prod_{i=1}^nb_{i,n+1-i} ∣Bn∣=(−1)2n(n−1)∏i=1nbi,n+1−i.

范德蒙: ∣ D n ∣ = ∏ 1 ≤ j ≤ i ≤ n ( x i − x j ) |D_n|=\prod_{1\leq j\leq i\leq n}(x_i-x_j) ∣Dn∣=∏1≤j≤i≤n(xi−xj).

矩阵: n × n n\times n n×n 方阵构成有零因子非交换环; 单位元 ∣ E ∣ = 1 |E|=1 ∣E∣=1; 零因子 ∣ Z ∣ = 0 |Z|=0 ∣Z∣=0.

反对称: a i j + a j i = 0    ⟹    a i i = 0 a_{ij}+a_{ji}=0\implies a_{ii}=0 aij+aji=0⟹aii=0.

转置: a i j T = a j i a^T_{ij}=a_{ji} aijT=aji; ( A + B ) T = A T + B T (A+B)^T=A^T+B^T (A+B)T=AT+BT; ( A B ) T = B T A T (AB)^T=B^TA^T (AB)T=BTAT.

逆元: A A − 1 = A − 1 A = E    ⟺    ∣ A ∣ ≠ 0 AA^{-1}=A^{-1}A=E\iff |A|\ne 0 AA−1=A−1A=E⟺∣A∣=0; ∣ A − 1 ∣ = ∣ A ∣ − 1 |A^{-1}|=|A|^{-1} ∣A−1∣=∣A∣−1; A − 1 = ∣ A ∣ − 1 A ∗ A^{-1}=|A|^{-1}A^* A−1=∣A∣−1A∗; ( A B ) − 1 = B − 1 A − 1 (AB)^{-1}=B^{-1}A^{-1} (AB)−1=B−1A−1.

A ∣ E \] → ∏ i = 1 k Q k \[ E ∣ A − 1 \] \[A\|E\]\\xrightarrow\[\]{\\prod_{i=1}\^kQ_k}\[E\|A\^{-1}\] \[A∣E\]∏i=1kQk \[E∣A−1\]. 伴随: A A ∗ = A ∗ A = ∣ A ∣ E AA\^\*=A\^\*A=\|A\|E AA∗=A∗A=∣A∣E; ∣ A ∗ ∣ = ∣ A ∣ n − 1 \|A\^\*\|=\|A\|\^{n-1} ∣A∗∣=∣A∣n−1; A ∗ = ∣ A ∣ A − 1 A\^\*=\|A\|A\^{-1} A∗=∣A∣A−1; ( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 2 ) (A\^\*)\^\*=\|A\|\^{n-2}A\\ (n\\geq 2) (A∗)∗=∣A∣n−2A (n≥2). 正交: A A T = A T A = E    ⟹    ∣ A ∣ = ± 1 AA\^T=A\^TA=E\\implies \|A\|=\\pm 1 AAT=ATA=E⟹∣A∣=±1, 行(列)向量均为单位向量并两两正交. 初等变换: 对换( E i j E_{ij} Eij); 倍乘( E i ( k ) , k ≠ 0 E_i(k),\\ k\\ne 0 Ei(k), k=0); 倍加( E i j ( K ) E_{ij}(K) Eij(K)); 左乘为行变换, 右乘为列变换. ∣ E i j A ∣ = ∣ A E i j ∣ = − ∣ A ∣ \|E_{ij}A\|=\|AE_{ij}\|=-\|A\| ∣EijA∣=∣AEij∣=−∣A∣; ∣ E i ( k ) A ∣ = ∣ A E i ( k ) ∣ = k ∣ A ∣ \|E_i(k)A\|=\|AE_i(k)\|=k\|A\| ∣Ei(k)A∣=∣AEi(k)∣=k∣A∣; ∣ k A ∣ = k n ∣ A ∣ \|kA\|=k\^n\|A\| ∣kA∣=kn∣A∣; ∣ E i j ( k ) A ∣ = ∣ A E i j ( k ) ∣ = ∣ A ∣ \|E_{ij}(k)A\|=\|AE_{ij}(k)\|=\|A\| ∣Eij(k)A∣=∣AEij(k)∣=∣A∣. 等价: A ≅ B    ⟺    ∃ ∣ P ∣ , ∣ Q ∣ ≠ 0 A\\cong B\\iff\\exists \|P\|,\|Q\|\\ne 0 A≅B⟺∃∣P∣,∣Q∣=0 s.t. P A Q = B PAQ=B PAQ=B. 秩: 非零子式最大阶数; 极大线性无关组中向量个数. r ( A ) = 0    ⟺    A = O r(A)=0\\iff A=O r(A)=0⟺A=O. r ( A ) = 1    ⟺    A ≠ O r(A)=1\\iff A\\ne O r(A)=1⟺A=O 且任意两行(列)成比例. r ( A A T ) = r ( A T A ) = r ( A ) r(AA\^T)=r(A\^TA)=r(A) r(AAT)=r(ATA)=r(A). r ( A n ) \< n    ⟺    ∣ A n ∣ = 0    ⟺    r(A_n)\ r ( B )    ⟹    r(A)=r(A\|B)\>r(B)\\implies r(A)=r(A∣B)\>r(B)⟹ 向量组 B B B 可由向量组 A A A 线性表出, 向量组 A A A 不可由向量组 B B B 线性表出. 向量组 A , B A,B A,B 可互相线性表出    ⟺    A ≅ B    ⟺    r ( A ) = r ( A ∣ B ) = r ( B ) \\iff A\\cong B\\iff r(A)=r(A\|B)=r(B) ⟺A≅B⟺r(A)=r(A∣B)=r(B). 线性无关: { α i } i = 1 m \\{\\bm{\\alpha}_i\\}_{i=1}\^m {αi}i=1m, 仅当 { k i } i = 1 m = 0 \\{k_i\\}_{i=1}\^m=0 {ki}i=1m=0 时才有 ∑ i = 1 m k i α i = 0    ⟺    r ( α i ) i = 1 m = m    ⟺    ( α i ) i = 1 m x = 0 \\sum_{i=1}\^m k_i\\bm{\\alpha}_i=\\bm{0}\\iff r(\\bm{\\alpha}_i)_{i=1}\^m=m\\iff (\\bm{\\alpha}_i)_{i=1}\^m\\bm{x}=\\bm{0} ∑i=1mkiαi=0⟺r(αi)i=1m=m⟺(αi)i=1mx=0 仅有零解. 全体组线性无关    ⟹    \\implies ⟹ 部分组线性无关. 缩短组线性无关    ⟹    \\implies ⟹ 延伸组线性无关. 向量组 B t B_t Bt 线性无关, 可由向量组 A s A_s As 线性表出    ⟹    s ≥ t \\implies s\\geq t ⟹s≥t. 向量组 A A A 线性无关, 添加 β \\bm{\\beta} β 后线性相关    ⟹    β \\implies\\bm{\\beta} ⟹β 可由向量组 A A A 线性表出且方式唯一. v = A x = B y \\bm{v}=A\\bm{x}=B\\bm{y} v=Ax=By. 过渡矩阵: 基 A A A 变换为基 B B B, 有 B = A P B=AP B=AP. 坐标变换: 坐标 x \\bm{x} x 变换为 y \\bm{y} y, 即 y = P − 1 x \\bm{y}=P\^{-1}\\bm{x} y=P−1x. 施密特正交化: β 1 = α 1 \\bm{\\beta}_1=\\bm{\\alpha}_1 β1=α1; β i = α i − ∑ j = 1 r − 1 \[ α i , β j \] \[ β j , β j \] , i = 2 , 3 , . . . , r \\bm{\\beta}_i=\\bm{\\alpha}_i-\\sum_{j=1}\^{r-1}\\frac{\[\\bm{\\alpha}_i,\\bm{\\beta}_j\]}{\[\\bm{\\beta}_j,\\bm{\\beta}_j\]},\\ i=2,3,...,r βi=αi−∑j=1r−1\[βj,βj\]\[αi,βj\], i=2,3,...,r. 单位化: γ i = β i ∣ ∣ β i ∣ ∣ , i = 1 , 2 , . . . , r \\bm{\\gamma}_i=\\frac{\\bm{\\beta}_i}{\|\|\\bm{\\beta}_i\|\|},\\ i=1,2,...,r γi=∣∣βi∣∣βi, i=1,2,...,r. A x = 0 A\\bm{x}=\\bm{0} Ax=0 仅有零解(唯一解)    ⟺    r ( A ) = n \\iff r(A)=n ⟺r(A)=n. A x = 0 A\\bm{x}=\\bm{0} Ax=0 有非零解(无穷解)    ⟺    r ( A ) \< n \\iff r(A)\ 0 \\bm{x}\^TA\\bm{x}\>0 xTAx\>0. A n A_n An 正定    ⟺    \\iff ⟺ 正惯性指数为 n    ⟺    n\\iff n⟺ 标准形平方项系数全为正    ⟺    \\iff ⟺ 特征值均大于 0    ⟺    0\\iff 0⟺ 各阶顺序主子式均大于 0    ⟺    ∃ ∣ P ∣ ≠ 0 0\\iff\\exists \|P\|\\ne 0 0⟺∃∣P∣=0 s.t. A = P T P A=P\^TP A=PTP, 即 A ≃ E    ⟹    a i i \> 0 , i = 1 , 2 , . . . , n A\\simeq E\\implies a_ii\>0,\\ i=1,2,...,n A≃E⟹aii\>0, i=1,2,...,n, ∣ A ∣ \> 0 \|A\|\>0 ∣A∣\>0.