∣ 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)<n\iff|A_n|=0\iff r(An)<n⟺∣An∣=0⟺ 不可逆 ⟺ \iff ⟺ 有零特征值.
r ( A n ) = n ⟺ ∣ A n ∣ ≠ 0 ⟺ r(A_n)=n\iff|A_n|\ne 0\iff r(An)=n⟺∣An∣=0⟺ 可逆 ⟺ \iff ⟺ 无零特征值.
r ( A ± B ) ≤ r ( A ) + r ( B ) r(A\pm B)\leq r(A)+r(B) r(A±B)≤r(A)+r(B); r ( A B ) ≤ min { r ( A ) , r ( B ) } r(AB)\leq\min\{r(A),r(B)\} r(AB)≤min{r(A),r(B)}; max { r ( A ) , r ( B ) } ≤ r ( A ∣ B ) ≤ r ( A ) + r ( B ) \max\{r(A),r(B)\}\leq r(A|B)\leq r(A)+r(B) max{r(A),r(B)}≤r(A∣B)≤r(A)+r(B).
r ( A m × n ) = n ⟹ r ( A B ) = r ( B ) r(A_{m\times n})=n\implies r(AB)=r(B) r(Am×n)=n⟹r(AB)=r(B).
r ( C n × s ) = n ⟹ r ( B C ) = r ( B ) r(C_{n\times s})=n\implies r(BC)=r(B) r(Cn×s)=n⟹r(BC)=r(B).
A ≅ B ⟹ r ( A ) = r ( B ) A\cong B\implies r(A)=r(B) A≅B⟹r(A)=r(B).
A m × n B n × s = O ⟹ r ( A ) + r ( B ) ≤ n A_{m\times n}B_{n\times s}=O\implies r(A)+r(B)\leq n Am×nBn×s=O⟹r(A)+r(B)≤n.
r ( A ∗ ) = { n , r ( A ) = n 1 , r ( A ) = n − 1 0 , r ( A ) < n − 1 r(A^*)=\begin{cases}n, & r(A)=n \\ 1, & r(A)=n-1 \\ 0, & r(A)<n-1\end{cases} r(A∗)=⎩ ⎨ ⎧n,1,0,r(A)=nr(A)=n−1r(A)<n−1.
α , β ≠ 0 \bm{\alpha},\bm{\beta}\ne\bm{0} α,β=0 为等维列向量 ⟹ r ( α β T ) = 1 \implies r(\bm{\alpha}\bm{\beta}^T)=1 ⟹r(αβT)=1.
线性表出: { α i } i = 1 m \{\bm{\alpha}i\}{i=1}^m {αi}i=1m, β \bm{\beta} β, ∃ { k i } i = 1 m \exists \{k_i\}{i=1}^m ∃{ki}i=1m s.t. β = ∑ i = 1 m k i α i ⟺ r ( α i ) i = 1 m = r ( { α i } i = 1 m ∣ β ) ⟺ ( α i ) i = 1 m x = β \bm{\beta}=\sum{i=1}^m k_i\bm{\alpha}_i\iff r(\bm{\alpha}i){i=1}^m=r(\{\bm{\alpha}i\}{i=1}^m|\beta)\iff(\bm{\alpha}i){i=1}^m\bm{x}=\bm{\beta} β=∑i=1mkiαi⟺r(αi)i=1m=r({αi}i=1m∣β)⟺(αi)i=1mx=β 有解.
r ( A ) = r ( A ∣ B ) > 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−1βj,βjαi,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)<n ⟺r(A)<n.
A x = b A\bm{x}=\bm{b} Ax=b 有唯一解 ⟺ r ( A ) = r ( A ∣ b ) = n \iff r(A)=r(A|\bm{b})=n ⟺r(A)=r(A∣b)=n.
A x = b A\bm{x}=\bm{b} Ax=b 有无穷解 ⟺ r ( A ) = r ( A ∣ b ) < n \iff r(A)=r(A|\bm{b})<n ⟺r(A)=r(A∣b)<n.
A x = b A\bm{x}=\bm{b} Ax=b 无解 ⟺ r ( A ) < r ( A ∣ b ) \iff r(A)<r(A|\bm{b}) ⟺r(A)<r(A∣b), 即 r ( A ) = r ( A ∣ b ) − 1 r(A)=r(A|\bm{b})-1 r(A)=r(A∣b)−1.
基础解系: A x = 0 A\bm{x}=\bm{0} Ax=0 的 s = n − r ( A ) s=n-r(A) s=n−r(A) 个线性无关解 { ξ i } i = 1 s \{\bm{\xi}i\}{i=1}^s {ξi}i=1s; 通解 x = ∑ i = 1 s k i ξ i \bm{x}=\sum_{i=1}^sk_i\bm{\xi}_i x=∑i=1skiξi.
A x = 0 A\bm{x}=\bm{0} Ax=0 的解的线性组合也为 A x = 0 A\bm{x}=0 Ax=0 的解.
ξ \bm{\xi} ξ 为 A x = 0 A\bm{x}=0 Ax=0 的解, η {\eta} η 为 A x = b A\bm{x}=\bm{b} Ax=b 的解 ⟹ η + k ξ \implies\bm{\eta}+k\bm{\xi} ⟹η+kξ 为 A x = b A\bm{x}=\bm{b} Ax=b 的解.
A x = b A\bm{x}=\bm{b} Ax=b 的解的系数和为 0 0 0 的线性组合为 A x = 0 A\bm{x}=\bm{0} Ax=0 的解; 系数和为 1 1 1 的线性组合为 A x = b A\bm{x}=\bm{b} Ax=b 的解.
A x = 0 A\bm{x}=\bm{0} Ax=0 和 B x = 0 B\bm{x}=\bm{0} Bx=0 有非零公共解 ⟹ r ( A B ) < n \implies r{A\choose B}<n ⟹r(BA)<n.
A x = 0 A\bm{x}=\bm{0} Ax=0 的解均为 B x = 0 B\bm{x}=\bm{0} Bx=0 的解 ⟹ r ( A ) ≥ r ( B ) \implies r(A)\geq r(B) ⟹r(A)≥r(B).
A x = 0 A\bm{x}=\bm{0} Ax=0 和 B x = 0 B\bm{x}=\bm{0} Bx=0 同解 ⟹ r ( A ) = r ( B ) = r ( A B ) \implies r(A)=r(B)=r{A\choose B} ⟹r(A)=r(B)=r(BA).
A T A x = 0 A^TA\bm{x}=\bm{0} ATAx=0 和 A x = 0 A\bm{x}=\bm{0} Ax=0 同解.
A B x = 0 AB\bm{x}=\bm{0} ABx=0 和 B x = 0 B\bm{x}=\bm{0} Bx=0 同解; A A A 为列满秩.
A n n x = 0 A_n^n\bm{x}=\bm{0} Annx=0 和 A n n + 1 x = 0 A_n^{n+1}\bm{x}=\bm{0} Ann+1x=0 同解.
特征值: A α = λ α A\bm{\alpha}=\lambda\bm{\alpha} Aα=λα; ∣ A − λ E ∣ = 0 |A-\lambda E|=0 ∣A−λE∣=0.
正交矩阵特征值为 ± 1 \pm 1 ±1.
f ( A ) = 0 ⟹ f ( λ ) = 0 f(A)=0\implies f(\lambda)=0 f(A)=0⟹f(λ)=0.
特征向量: ( A − λ E ) x = 0 (A-\lambda E)\bm{x}=0 (A−λE)x=0.
k k k 重特征值至多有 k k k 个线性无关的特征向量; 不同特征值对应的特征向量线性无关; 一个特征向量对应一个特征值; 全部线性无关的特征向量的线性组合也为该特征值的特征向量.
| 矩阵 | 特征值 | 特征向量 |
|---|---|---|
| A A A | λ \lambda λ | α \bm{\alpha} α |
| A + k E A+kE A+kE | λ + k \lambda+k λ+k | α \bm{\alpha} α |
| k A kA kA | k λ k\lambda kλ | α \bm{\alpha} α |
| A k A^k Ak | λ k \lambda^k λk | α \bm{\alpha} α |
| f ( A ) f(A) f(A) | f ( λ ) f(\lambda) f(λ) | α \bm{\alpha} α |
| A − 1 A^{-1} A−1 | 1 λ \frac{1}{\lambda} λ1 | α \bm{\alpha} α |
| A ∗ A^* A∗ | ∣ A ∣ λ \frac{|A|}{\lambda} λ∣A∣ | α \bm{\alpha} α |
| A T A^T AT | λ \lambda λ | - |
| P − 1 A P P^{-1}AP P−1AP | λ \lambda λ | P − 1 α P^{-1}\bm{\alpha} P−1α |
相似: A ∼ B ⟺ ∃ ∣ P ∣ ≠ 0 A\sim B\iff\exists|P|\ne 0 A∼B⟺∃∣P∣=0 s.t. P − 1 A P = B P^{-1}AP=B P−1AP=B.
A ∼ B ⟹ ∣ A − λ E ∣ = ∣ B − λ E ∣ ⟹ A ≅ B A\sim B\implies |A-\lambda E|=|B-\lambda E|\implies A\cong B A∼B⟹∣A−λE∣=∣B−λE∣⟹A≅B, t r ( A ) = t r ( B ) {\rm tr}(A)={\rm tr}(B) tr(A)=tr(B), ∣ A ∣ = ∣ B ∣ |A|=|B| ∣A∣=∣B∣, 各阶主子式之和分别相等.
A n ∼ Λ ⟺ A A_n\sim\Lambda\iff A An∼Λ⟺A 有 n n n 个线性无关的特征向量, 此时 P = ( α i ) i = 1 n P=(\bm{\alpha}i){i=1}^n P=(αi)i=1n, Λ = P − 1 A P ⟺ A \Lambda=P^{-1}AP\iff A Λ=P−1AP⟺A 的 k k k 重特征根恰有 k k k 个线性无关的特征向量, 即 r ( A − λ E ) = n − k r(A-\lambda E)=n-k r(A−λE)=n−k.
A n A_n An 有 n n n 个不同的特征值 ⟹ A ∼ Λ \implies A\sim\Lambda ⟹A∼Λ.
( A − λ 1 E ) ( A − λ 2 E ) = O , λ 1 ≠ λ 2 ⟹ A ∼ Λ (A-\lambda_1 E)(A-\lambda_2 E)=O,\ \lambda_1\ne\lambda_2\implies A\sim\Lambda (A−λ1E)(A−λ2E)=O, λ1=λ2⟹A∼Λ.
A = P − 1 Λ P ⟹ A k = P − 1 Λ k P A=P^{-1}\Lambda P\implies A^k=P^{-1}\Lambda^k P A=P−1ΛP⟹Ak=P−1ΛkP.
A = ( λ E + B ) ⟹ A k = ∑ i = 1 k ( k i ) λ i B k − i A=(\lambda E+B)\implies A^k=\sum_{i=1}^k{k\choose i}\lambda^iB^{k-i} A=(λE+B)⟹Ak=∑i=1k(ik)λiBk−i; B j = O B^j=O Bj=O, j ≪ k j\ll k j≪k.
A = α β T ⟹ A k = c k − 1 A A=\bm{\alpha}\bm{\beta}^T\implies A^k=c^{k-1}A A=αβT⟹Ak=ck−1A, c = β T α c=\bm{\beta}^T\bm{\alpha} c=βTα.
λ n = ∣ λ E − A ∣ Q ( λ ) + R ( λ ) ⟹ A n = R ( A ) \lambda^n=|\lambda E-A|Q(\lambda)+R(\lambda)\implies A^n=R(A) λn=∣λE−A∣Q(λ)+R(λ)⟹An=R(A).
A A A 为实对称矩阵: a i j = a j i ∈ R a_{ij}=a_{ji}\in\mathbb{R} aij=aji∈R;
特征值均为实数;
不同特征值的特征向量正交;
∃ \exists ∃ 正交矩阵 Q Q Q s.t. Q − 1 A Q = Q T A Q = Λ Q^{-1}AQ=Q^TAQ=\Lambda Q−1AQ=QTAQ=Λ, 即可正交相似对角化;
B B B 也为实对称矩阵, A ∼ B ⟹ { λ A } = { λ B } A\sim B\implies \{\lambda_A\}=\{\lambda_B\} A∼B⟹{λA}={λB}.
二次型: ∑ i = 1 n ∑ j = 1 n a i j x i x j = x T A x \sum_{i=1}^n\sum_{j=1}^na_{ij}x_ix_j=\bm{x}^TA\bm{x} ∑i=1n∑j=1naijxixj=xTAx; A A A 为实对称矩阵.
标准形: 只含平方项.
规范形: 只含平方项, 系数只能为 0 , 1 , − 1 0,1,-1 0,1,−1.
正交变换: x = Q y \bm{x}=Q\bm{y} x=Qy, 即 Q T A Q = Λ Q^TAQ=\Lambda QTAQ=Λ; 正交矩阵 Q = ( γ i = 1 n ) Q=(\bm{\gamma}_{i=1}^n) Q=(γi=1n) 为 A A A 特征向量的正交单位化.
配方换元: x = C y \bm{x}=C\bm{y} x=Cy, C C C 为可逆矩阵.
惯性定理: 标准形正负系数个数(正负惯性指数)分别等于正负特征值个数, 数量和为矩阵秩.
合同: A ≃ B ⟺ ∃ ∣ C ∣ ≠ 0 A\simeq B\iff\exists |C|\ne 0 A≃B⟺∃∣C∣=0 s.t. B = C T A C ⟺ A , B B=C^TAC\iff A,B B=CTAC⟺A,B 的正负惯性指数分别相等.
正定: ∀ x ≠ O \forall \bm{x}\ne O ∀x=O s.t. x T A x > 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.