文章目录
1 平面几何
1.1 三角形
三角形的三个顶点为 P A = ( x A , y A , z A ) P_A=(x_A,y_A,z_A) PA=(xA,yA,zA)、 P B = ( x B , y B , z B ) P_B=(x_B,y_B,z_B) PB=(xB,yB,zB)、 P C = ( x C , y C , z C ) P_C=(x_C,y_C,z_C) PC=(xC,yC,zC)，有如下：
a ⃗ = B C → = P C − P B b ⃗ = C A → = P A − P C c ⃗ = A B → = P B − P A \vec{a} = \overrightarrow{BC} = P_C-P_B \\ \vec{b} = \overrightarrow{CA} = P_A-P_C \\ \vec{c} = \overrightarrow{AB} = P_B-P_A a=BC=PC−PBb=CA=PA−PCc=AB=PB−PA
a = ∥ a ⃗ ∥ b = ∥ b ⃗ ∥ c = ∥ c ⃗ ∥ a = \lVert \vec{a} \rVert \\ b = \lVert \vec{b} \rVert \\ c = \lVert \vec{c} \rVert a=∥a∥b=∥b∥c=∥c∥
1.1.1 重心
定义：三条中线相交的点叫做重心。
P G = P A + P B + P C 3 P_G = \frac{P_A+P_B+P_C}{3} PG=3PA+PB+PC
1.1.2 外心
定义：三角形三边的垂直平分线的交点，称为三角形外心
d = 2 ∣ c ⃗ × a ⃗ ∣ 2 α = ( a ⃗ ⋅ a ⃗ ) ( c ⃗ ⋅ b ⃗ ) d β = ( b ⃗ ⋅ b ⃗ ) ( c ⃗ ⋅ a ⃗ ) d γ = ( c ⃗ ⋅ c ⃗ ) ( b ⃗ ⋅ a ⃗ ) d P = α P A + β P B + γ P C d = 2 \left | \vec{c} \times \vec{a} \right | ^2 \\ \alpha = \frac{\left( \vec{a} \cdot \vec{a} \right) \left( \vec{c} \cdot \vec{b} \right)}{d} \\ \beta = \frac{\left( \vec{b} \cdot \vec{b} \right) \left( \vec{c} \cdot \vec{a} \right)}{d} \\ \gamma = \frac{\left( \vec{c} \cdot \vec{c} \right) \left( \vec{b} \cdot \vec{a} \right)}{d} \\ P = \alpha P_A + \beta P_B + \gamma P_C d=2∣c×a∣2α=d(a⋅a)(c⋅b)β=d(b⋅b)(c⋅a)γ=d(c⋅c)(b⋅a)P=αPA+βPB+γPC
1.1.3 内心
定义：三角形内心为三角形三条内角平分线的交点。
P = a P A + b P B + c P C a + b + c P = \frac{aP_A+bP_B+cP_C}{a+b+c} P=a+b+caPA+bPB+cPC
1.1.4 垂心
定义：三角形三边上的三条高或其延长线交于一点，称为三角形垂心
α = ( a ⃗ ⋅ b ⃗ ) ( a ⃗ ⋅ c ⃗ ) β = ( b ⃗ ⋅ c ⃗ ) ( b ⃗ ⋅ a ⃗ ) γ = ( c ⃗ ⋅ a ⃗ ) ( c ⃗ ⋅ b ⃗ ) P = α P A + β P B + γ P C α + β + γ \alpha = \left( \vec{a} \cdot \vec{b} \right) \left( \vec{a} \cdot \vec{c} \right) \\ \beta = \left( \vec{b} \cdot \vec{c} \right) \left( \vec{b} \cdot \vec{a} \right) \\ \gamma = \left( \vec{c} \cdot \vec{a} \right) \left( \vec{c} \cdot \vec{b} \right) \\ P = \frac{\alpha P_A + \beta P_B + \gamma P_C}{\alpha + \beta + \gamma} α=(a⋅b)(a⋅c)β=(b⋅c)(b⋅a)γ=(c⋅a)(c⋅b)P=α+β+γαPA+βPB+γPC
1.1.5 旁心
定义：与三角形的一边及其他两边的延长线都相切的圆叫做三角形的旁切圆，旁切圆的圆心叫做三角形旁心
2 矢量
2.1 矢量运算
2.1.1 矢量三重积
a ⃗ × ( b ⃗ × c ⃗ ) = b ⃗ ( a ⃗ ⋅ c ⃗ ) − c ⃗ ( a ⃗ ⋅ b ⃗ ) \vec{a} \times \left(\vec{b} \times \vec{c} \right) = \vec{b} \left(\vec{a} \cdot \vec{c} \right) - \vec{c} \left(\vec{a} \cdot \vec{b} \right) a×(b×c)=b(a⋅c)−c(a⋅b)
3 三角函数
3.1 倍角公式
sin 2 α = 2 sin α cos α \sin{2\alpha} = 2\sin{\alpha}\cos{\alpha} sin2α=2sinαcosα
cos 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α = cos 2 α − sin 2 α = 1 − tan 2 α 1 + tan 2 α \cos{2\alpha} = 2\cos^2{\alpha} - 1 = 1 - 2\sin^2{\alpha} = \cos^2{\alpha} - \sin^2{\alpha} = \frac{1-\tan^2{\alpha}}{1+\tan^2{\alpha}} cos2α=2cos2α−1=1−2sin2α=cos2α−sin2α=1+tan2α1−tan2α
tan 2 α = 2 tan α 1 − tan 2 α = 2 cot α cot 2 α − 1 = 2 cot α − tan α \tan{2\alpha} = \frac{2\tan{\alpha}}{1-\tan^2{\alpha}} = \frac{2\cot{\alpha}}{\cot^2{\alpha}-1} = \frac{2}{\cot{\alpha}-\tan{\alpha}} tan2α=1−tan2α2tanα=cot2α−12cotα=cotα−tanα2
cot 2 α = cot 2 α − 1 2 cot α \cot{2\alpha} = \frac{\cot^2{\alpha}-1}{2\cot{\alpha}} cot2α=2cotαcot2α−1
sec 2 α = sec 2 α + csc 2 α csc 2 α − sec 2 α = sec 2 α csc 2 α csc 2 α − sec 2 α \sec{2\alpha} = \frac{\sec^2{\alpha}+\csc^2{\alpha}}{\csc^2{\alpha}-\sec^2{\alpha}} = \frac{\sec^2{\alpha}\csc^2{\alpha}}{\csc^2{\alpha}-\sec^2{\alpha}} sec2α=csc2α−sec2αsec2α+csc2α=csc2α−sec2αsec2αcsc2α
csc 2 α = sec 2 α + csc 2 α 2 sec α csc α = sec 2 α csc 2 α 2 sec α csc α \csc{2\alpha} = \frac{\sec^2{\alpha}+\csc^2{\alpha}}{2\sec{\alpha}\csc{\alpha}} = \frac{\sec^2{\alpha}\csc^2{\alpha}}{2\sec{\alpha}\csc{\alpha}} csc2α=2secαcscαsec2α+csc2α=2secαcscαsec2αcsc2α
3.2 半角公式
sin α 2 = ± 1 − cos α 2 \sin{\frac{\alpha}{2}} = \pm \sqrt{\frac{1-\cos{\alpha}}{2}} sin2α=±21−cosα
cos α 2 = ± 1 + cos α 2 \cos{\frac{\alpha}{2}} = \pm \sqrt{\frac{1+\cos{\alpha}}{2}} cos2α=±21+cosα
tan α 2 = sin α 1 + cos α = 1 − cos α sin α = ± 1 − cos α 1 + cos α \tan{\frac{\alpha}{2}} = \frac{\sin{\alpha}}{1+\cos{\alpha}} = \frac{1-\cos{\alpha}}{\sin{\alpha}} = \pm \sqrt{\frac{1-\cos{\alpha}}{1+\cos{\alpha}}} tan2α=1+cosαsinα=sinα1−cosα=±1+cosα1−cosα
3.3 和角公式
sin ( α + β ) = sin α cos β + cos α sin β \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta sin(α+β)=sinαcosβ+cosαsinβ
sin ( α − β ) = sin α cos β − cos α sin β \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta sin(α−β)=sinαcosβ−cosαsinβ
cos ( α + β ) = cos α cos β − sin α sin β \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta cos(α+β)=cosαcosβ−sinαsinβ
cos ( α − β ) = cos α cos β + sin α sin β \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta cos(α−β)=cosαcosβ+sinαsinβ
3.4 和差化积
sin α + sin β = 2 sin α + β 2 cos α − β 2 \sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} sinα+sinβ=2sin2α+βcos2α−β
sin α − sin β = 2 cos α + β 2 sin α − β 2 \sin \alpha - \sin \beta = 2 \cos \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} sinα−sinβ=2cos2α+βsin2α−β
cos α + cos β = 2 cos α + β 2 cos α − β 2 \cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2} cosα+cosβ=2cos2α+βcos2α−β
cos α − cos β = − 2 sin α + β 2 sin α − β 2 \cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2} cosα−cosβ=−2sin2α+βsin2α−β
tan α + tan β = sin ( α + β ) cos α cos β \tan \alpha + \tan \beta = \frac{\sin (\alpha + \beta)}{\cos \alpha \cos \beta} tanα+tanβ=cosαcosβsin(α+β)
tan α − tan β = sin ( α − β ) cos α cos β \tan \alpha - \tan \beta = \frac{\sin (\alpha - \beta)}{\cos \alpha \cos \beta} tanα−tanβ=cosαcosβsin(α−β)
cot α + cot β = sin ( α + β ) sin α sin β \cot \alpha + \cot \beta = \frac{\sin (\alpha + \beta)}{\sin \alpha \sin \beta} cotα+cotβ=sinαsinβsin(α+β)
cot α − cot β = − sin ( α − β ) sin α sin β \cot \alpha - \cot \beta = -\frac{\sin (\alpha - \beta)}{\sin \alpha \sin \beta} cotα−cotβ=−sinαsinβsin(α−β)
tan α + cot β = cos ( α − β ) cos α sin β \tan \alpha + \cot \beta = \frac{\cos (\alpha - \beta)}{\cos \alpha \sin \beta} tanα+cotβ=cosαsinβcos(α−β)
tan α − cot β = − cos ( α + β ) cos α sin β \tan \alpha - \cot \beta = -\frac{\cos (\alpha + \beta)}{\cos \alpha \sin \beta} tanα−cotβ=−cosαsinβcos(α+β)
sin 2 α − sin 2 β = sin ( α + β ) sin ( α − β ) \sin^2 \alpha - \sin^2 \beta = \sin (\alpha + \beta) \sin (\alpha - \beta) sin2α−sin2β=sin(α+β)sin(α−β)
cos 2 α − cos 2 β = − sin ( α + β ) sin ( α − β ) \cos^2 \alpha - \cos^2 \beta = -\sin (\alpha + \beta) \sin (\alpha - \beta) cos2α−cos2β=−sin(α+β)sin(α−β)
3.5 积化和差
sin α cos β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] \sin \alpha \cos \beta = \frac{1}{2} \left[ \sin (\alpha + \beta) + \sin (\alpha - \beta) \right] sinαcosβ=21[sin(α+β)+sin(α−β)]
cos α sin β = 1 2 [ sin ( α + β ) − sin ( α − β ) ] \cos \alpha \sin \beta = \frac{1}{2} \left[ \sin (\alpha + \beta) - \sin (\alpha - \beta) \right] cosαsinβ=21[sin(α+β)−sin(α−β)]
cos α cos β = 1 2 [ cos ( α + β ) + cos ( α − β ) ] \cos \alpha \cos \beta = \frac{1}{2} \left[ \cos (\alpha + \beta) + \cos (\alpha - \beta) \right] cosαcosβ=21[cos(α+β)+cos(α−β)]
sin α sin β = − 1 2 [ cos ( α + β ) − cos ( α − β ) ] \sin \alpha \sin \beta = -\frac{1}{2} \left[ \cos (\alpha + \beta) - \cos (\alpha - \beta) \right] sinαsinβ=−21[cos(α+β)−cos(α−β)]
4 矩阵
4.1 分块矩阵
4.1.1 行列式
∣ A B ∣ = ∣ A ∣ ⋅ ∣ B ∣ \lvert AB \rvert = \lvert A \rvert \cdot \lvert B \rvert ∣AB∣=∣A∣⋅∣B∣
注： A A A、 B B B为方阵
- 情况1
[ A 1 , 1 A 1 , 2 A 2 , 1 A 2 , 2 ] [ B 1 , 1 B 1 , 2 B 2 , 1 B 2 , 2 ] = [ A 1 , 1 B 1 , 1 + A 1 , 2 B 2 , 1 A 1 , 1 B 1 , 2 + A 1 , 2 B 2 , 2 A 2 , 1 B 1 , 1 + A 2 , 2 B 2 , 1 A 2 , 1 B 1 , 2 + A 2 , 2 B 2 , 2 ] \begin{bmatrix} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2,2} \end{bmatrix} \begin{bmatrix} B_{1,1} & B_{1,2} \\ B_{2,1} & B_{2,2} \end{bmatrix} = \begin{bmatrix} A_{1,1}B_{1,1}+A_{1,2}B_{2,1} & A_{1,1}B_{1,2}+A_{1,2}B_{2,2} \\ A_{2,1}B_{1,1}+A_{2,2}B_{2,1} & A_{2,1}B_{1,2}+A_{2,2}B_{2,2} \end{bmatrix} [A1,1A2,1A1,2A2,2][B1,1B2,1B1,2B2,2]=[A1,1B1,1+A1,2B2,1A2,1B1,1+A2,2B2,1A1,1B1,2+A1,2B2,2A2,1B1,2+A2,2B2,2]
- 情况2
∣ A 1 0 ⋯ 0 0 A 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n ∣ = ∣ A 1 ∣ ∣ A 2 ∣ … ∣ A n ∣ \begin{vmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & A_n \end{vmatrix} = \lvert A_1 \rvert \lvert A_2 \rvert \dots \lvert A_n \rvert A10⋮00A2⋮0⋯⋯⋱⋯00⋮An =∣A1∣∣A2∣…∣An∣
- 情况3
∣ A 0 C D ∣ = ∣ A ∣ ⋅ ∣ D ∣ \begin{vmatrix} A & 0 \\ C & D \end{vmatrix} = \lvert A \rvert \cdot \lvert D \rvert AC0D =∣A∣⋅∣D∣
注： A A A、 D D D为方阵
- 情况4
∣ A B 0 D ∣ = ∣ A ∣ ⋅ ∣ D ∣ \begin{vmatrix} A & B \\ 0 & D \end{vmatrix} = \lvert A \rvert \cdot \lvert D \rvert A0BD =∣A∣⋅∣D∣
注： A A A、 D D D为方阵
- 情况5
∣ 0 B C D ∣ = ∣ A B C 0 ∣ = ( − 1 ) m n ∣ B ∣ ⋅ ∣ C ∣ \begin{vmatrix} 0 & B \\ C & D \end{vmatrix} = \begin{vmatrix} A & B \\ C & 0 \end{vmatrix} = \left( -1 \right)^{mn} \lvert B \rvert \cdot \lvert C \rvert 0CBD = ACB0 =(−1)mn∣B∣⋅∣C∣
注： B B B、 C C C为方阵， D D D为 n × m n \times m n×m矩阵
- 情况6
∣ A B B A ∣ = ∣ A + B ∣ ⋅ ∣ A − B ∣ \begin{vmatrix} A & B \\ B & A \end{vmatrix} = \lvert A+B \rvert \cdot \lvert A-B \rvert ABBA =∣A+B∣⋅∣A−B∣
注： A A A、 B B B为方阵
- 情况7
∣ A − B B A ∣ = ∣ A + i B ∣ ⋅ ∣ A − i B ∣ \begin{vmatrix} A & -B \\ B & A \end{vmatrix} = \lvert A+iB \rvert \cdot \lvert A-iB \rvert AB−BA =∣A+iB∣⋅∣A−iB∣
注： A A A、 B B B为方阵
- 情况8
∣ A B C D ∣ = ∣ A ∣ ⋅ ∣ D − C A − 1 B ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert A \rvert \cdot \lvert D-CA^{-1}B \rvert ACBD =∣A∣⋅∣D−CA−1B∣
注： A A A可逆
- 情况9
∣ A B C D ∣ = ∣ D ∣ ⋅ ∣ A − B D − 1 C ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert D \rvert \cdot \lvert A-BD^{-1}C \rvert ACBD =∣D∣⋅∣A−BD−1C∣
注： D D D可逆
- 情况10
∣ A B C D ∣ = ∣ A D − C B ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert AD-CB \rvert ACBD =∣AD−CB∣
注： A C = C A AC=CA AC=CA
- 情况11
∣ A B C D ∣ = ∣ A D − B C ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert AD-BC \rvert ACBD =∣AD−BC∣
注： C D = D C CD=DC CD=DC
- 情况12
∣ A B C D ∣ = ∣ D A − B C ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert DA-BC \rvert ACBD =∣DA−BC∣
注： B D = D B BD=DB BD=DB
- 情况13
∣ A B C D ∣ = ∣ D A − C B ∣ \begin{vmatrix} A & B \\ C & D \end{vmatrix} = \lvert DA-CB \rvert ACBD =∣DA−CB∣
注： A B = B A AB=BA AB=BA
4.1.2 逆
- 情况1
设 A A A是 m × m m \times m m×m可逆矩阵， B B B是 m × n m \times n m×n矩阵， C C C是 n × m n \times m n×m矩阵， D D D是 n × n n \times n n×n矩阵， D − C A − 1 B D-CA^{-1}B D−CA−1B是 n × n n \times n n×n可逆矩阵，则有
[ A B C D ] − 1 = [ A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 1 ] \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} + A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\ -(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1} \end{bmatrix} [ACBD]−1=[A−1+A−1B(D−CA−1B)−1CA−1−(D−CA−1B)−1CA−1−A−1B(D−CA−1B)−1(D−CA−1B)−1]
当 A = I A = I A=I时：
[ I B C D ] − 1 = [ I + B ( D − C B ) − 1 C − B ( D − C B ) − 1 − ( D − C B ) − 1 C ( D − C B ) − 1 ] \begin{bmatrix} I & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} I + B(D-CB)^{-1}C & -B(D-CB)^{-1} \\ -(D-CB)^{-1}C & (D-CB)^{-1} \end{bmatrix} [ICBD]−1=[I+B(D−CB)−1C−(D−CB)−1C−B(D−CB)−1(D−CB)−1]
当 B = 0 B = 0 B=0时：
[ A 0 C D ] − 1 = [ A − 1 0 − D − 1 C A − 1 D − 1 ] \begin{bmatrix} A & 0 \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} & 0 \\ -D^{-1}CA^{-1} & D^{-1} \end{bmatrix} [AC0D]−1=[A−1−D−1CA−10D−1]
当 C = 0 C = 0 C=0时：
[ A B 0 D ] − 1 = [ A − 1 − A − 1 B D − 1 0 D − 1 ] \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} & -A^{-1}BD^{-1} \\ 0 & D^{-1} \end{bmatrix} [A0BD]−1=[A−10−A−1BD−1D−1]
当 D = 0 D = 0 D=0时：
[ A B C 0 ] − 1 = [ A − 1 − A − 1 B ( C A − 1 B ) − 1 C A − 1 A − 1 B ( C A − 1 B ) − 1 ( C A − 1 B ) − 1 C A − 1 − ( C A − 1 B ) − 1 ] \begin{bmatrix} A & B \\ C & 0 \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} - A^{-1}B(CA^{-1}B)^{-1}CA^{-1} & A^{-1}B(CA^{-1}B)^{-1} \\ (CA^{-1}B)^{-1}CA^{-1} & -(CA^{-1}B)^{-1} \end{bmatrix} [ACB0]−1=[A−1−A−1B(CA−1B)−1CA−1(CA−1B)−1CA−1A−1B(CA−1B)−1−(CA−1B)−1]
- 情况2
设 A A A是 m × n m \times n m×n矩阵， B B B是 m × m m \times m m×m可逆矩阵， C C C是 n × n n \times n n×n矩阵， D D D是 n × m n \times m n×m矩阵， C − D B − 1 A C-DB^{-1}A C−DB−1A是 n × n n \times n n×n可逆矩阵，则有
[ A B C D ] − 1 = [ − ( C − D B − 1 A ) − 1 D B − 1 ( C − D B − 1 A ) − 1 B − 1 + B − 1 A ( C − D B − 1 A ) − 1 D B − 1 − B − 1 A ( C − D B − 1 A ) − 1 ] \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -(C-DB^{-1}A)^{-1}DB^{-1} & (C-DB^{-1}A)^{-1} \\ B^{-1} + B^{-1}A(C-DB^{-1}A)^{-1}DB^{-1} & -B^{-1}A(C-DB^{-1}A)^{-1} \end{bmatrix} [ACBD]−1=[−(C−DB−1A)−1DB−1B−1+B−1A(C−DB−1A)−1DB−1(C−DB−1A)−1−B−1A(C−DB−1A)−1]
当 A = 0 A = 0 A=0时：
[ 0 B C D ] − 1 = [ − C − 1 D B − 1 C − 1 B − 1 0 ] \begin{bmatrix} 0 & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -C^{-1}DB^{-1} & C^{-1} \\ B^{-1} & 0 \end{bmatrix} [0CBD]−1=[−C−1DB−1B−1C−10]
当 B = I B = I B=I时：
[ A I C D ] − 1 = [ − ( C − D A ) − 1 D ( C − D A ) − 1 I + A ( C − D A ) − 1 D − A ( C − D A ) − 1 ] \begin{bmatrix} A & I \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -(C-DA)^{-1}D & (C-DA)^{-1} \\ I + A(C-DA)^{-1}D & -A(C-DA)^{-1} \end{bmatrix} [ACID]−1=[−(C−DA)−1DI+A(C−DA)−1D(C−DA)−1−A(C−DA)−1]
当 C = 0 C = 0 C=0时：
[ A B 0 D ] − 1 = [ ( D B − 1 A ) − 1 D B − 1 − ( D B − 1 A ) − 1 B − 1 − B − 1 A ( D B − 1 A ) − 1 D B − 1 B − 1 A ( D B − 1 A ) − 1 ] \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}^{-1} = \begin{bmatrix} (DB^{-1}A)^{-1}DB^{-1} & -(DB^{-1}A)^{-1} \\ B^{-1} - B^{-1}A(DB^{-1}A)^{-1}DB^{-1} & B^{-1}A(DB^{-1}A)^{-1} \end{bmatrix} [A0BD]−1=[(DB−1A)−1DB−1B−1−B−1A(DB−1A)−1DB−1−(DB−1A)−1B−1A(DB−1A)−1]
当 D = 0 D = 0 D=0时：
[ A B C 0 ] − 1 = [ 0 C − 1 B − 1 − B − 1 A C − 1 ] \begin{bmatrix} A & B \\ C & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & C^{-1} \\ B^{-1} & -B^{-1}AC^{-1} \end{bmatrix} [ACB0]−1=[0B−1C−1−B−1AC−1]
- 情况3
设 A A A是 m × n m \times n m×n矩阵， B B B是 m × m m \times m m×m矩阵， C C C是 n × n n \times n n×n可逆矩阵， B B B是 n × m n \times m n×m矩阵， B − A C − 1 D B-AC^{-1}D B−AC−1D是 m × m m \times m m×m可逆矩阵，则有
[ A B C D ] − 1 = [ − C − 1 D ( B − A C − 1 D ) − 1 C − 1 + C − 1 D ( B − A C − 1 D ) − 1 A C − 1 ( B − A C − 1 D ) − 1 − ( B − A C − 1 D ) − 1 A C − 1 ] \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -C^{-1}D(B-AC^{-1}D)^{-1} & C^{-1} + C^{-1}D(B-AC^{-1}D)^{-1}AC^{-1} \\ (B-AC^{-1}D)^{-1} & -(B-AC^{-1}D)^{-1}AC^{-1} \end{bmatrix} [ACBD]−1=[−C−1D(B−AC−1D)−1(B−AC−1D)−1C−1+C−1D(B−AC−1D)−1AC−1−(B−AC−1D)−1AC−1]
当 A = 0 A = 0 A=0时：
[ 0 B C D ] − 1 = [ − C − 1 D B − 1 C − 1 B − 1 0 ] \begin{bmatrix} 0 & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -C^{-1}DB^{-1} & C^{-1} \\ B^{-1} & 0 \end{bmatrix} [0CBD]−1=[−C−1DB−1B−1C−10]
当 B = 0 B = 0 B=0时：
[ A 0 C D ] − 1 = [ C − 1 D ( A C − 1 D ) − 1 C − 1 − C − 1 D ( A C − 1 D ) − 1 A C − 1 − ( A C − 1 D ) − 1 ( A C − 1 D ) − 1 A C − 1 ] \begin{bmatrix} A & 0 \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} C^{-1}D(AC^{-1}D)^{-1} & C^{-1} - C^{-1}D(AC^{-1}D)^{-1}AC^{-1} \\ -(AC^{-1}D)^{-1} & (AC^{-1}D)^{-1}AC^{-1} \end{bmatrix} [AC0D]−1=[C−1D(AC−1D)−1−(AC−1D)−1C−1−C−1D(AC−1D)−1AC−1(AC−1D)−1AC−1]
当 C = I C = I C=I时：
[ A B I D ] − 1 = [ − D ( B − A D ) − 1 I + D ( B − A D ) − 1 A ( B − A D ) − 1 − ( B − A D ) − 1 A ] \begin{bmatrix} A & B \\ I & D \end{bmatrix}^{-1} = \begin{bmatrix} -D(B-AD)^{-1} & I + D(B-AD)^{-1}A \\ (B-AD)^{-1} & -(B-AD)^{-1}A \end{bmatrix} [AIBD]−1=[−D(B−AD)−1(B−AD)−1I+D(B−AD)−1A−(B−AD)−1A]
当 D = 0 D = 0 D=0时：
[ A B C 0 ] − 1 = [ 0 C − 1 B − 1 − B − 1 A C − 1 ] \begin{bmatrix} A & B \\ C & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & C^{-1} \\ B^{-1} & -B^{-1}AC^{-1} \end{bmatrix} [ACB0]−1=[0B−1C−1−B−1AC−1]
- 情况4
设 A A A是 m × m m \times m m×m矩阵， B B B是 m × n m \times n m×n矩阵， C C C是 n × m n \times m n×m矩阵， D D D是 n × n n \times n n×n可逆矩阵， A − B D − 1 C A-BD^{-1}C A−BD−1C是 m × m m \times m m×m可逆矩阵，则有
[ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 − ( A − B D − 1 C ) − 1 B D − 1 − D − 1 C ( A − B D − 1 C ) − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 ] \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} (A-BD^{-1}C)^{-1} & -(A-BD^{-1}C)^{-1}BD^{-1} \\ -D^{-1}C(A-BD^{-1}C)^{-1} & D^{-1} + D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1} \end{bmatrix} [ACBD]−1=[(A−BD−1C)−1−D−1C(A−BD−1C)−1−(A−BD−1C)−1BD−1D−1+D−1C(A−BD−1C)−1BD−1]
当 A = 0 A = 0 A=0时：
[ 0 B C D ] − 1 = [ − ( B D − 1 C ) − 1 ( B D − 1 C ) − 1 B D − 1 D − 1 C ( B D − 1 C ) − 1 D − 1 − D − 1 C ( B D − 1 C ) − 1 B D − 1 ] \begin{bmatrix} 0 & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} -(BD^{-1}C)^{-1} & (BD^{-1}C)^{-1}BD^{-1} \\ D^{-1}C(BD^{-1}C)^{-1} & D^{-1} - D^{-1}C(BD^{-1}C)^{-1}BD^{-1} \end{bmatrix} [0CBD]−1=[−(BD−1C)−1D−1C(BD−1C)−1(BD−1C)−1BD−1D−1−D−1C(BD−1C)−1BD−1]
当 B = 0 B = 0 B=0时：
[ A 0 C D ] − 1 = [ A − 1 0 − D − 1 C A − 1 D − 1 ] \begin{bmatrix} A & 0 \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} & 0 \\ -D^{-1}CA^{-1} & D^{-1} \end{bmatrix} [AC0D]−1=[A−1−D−1CA−10D−1]
当 C = 0 C = 0 C=0时：
[ A B 0 D ] − 1 = [ A − 1 − A − 1 B D − 1 0 D − 1 ] \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} & -A^{-1}BD^{-1} \\ 0 & D^{-1} \end{bmatrix} [A0BD]−1=[A−10−A−1BD−1D−1]
当 D = I D = I D=I时：
[ A B C I ] − 1 = [ ( A − B C ) − 1 − ( A − B C ) − 1 B − C ( A − B C ) − 1 I + C ( A − B C ) − 1 B ] \begin{bmatrix} A & B \\ C & I \end{bmatrix}^{-1} = \begin{bmatrix} (A-BC)^{-1} & -(A-BC)^{-1}B \\ -C(A-BC)^{-1} & I + C(A-BC)^{-1}B \end{bmatrix} [ACBI]−1=[(A−BC)−1−C(A−BC)−1−(A−BC)−1BI+C(A−BC)−1B]
4.1.3 对角矩阵
A − 1 = [ A 1 0 ⋯ 0 0 A 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n ] − 1 = [ A 1 − 1 0 ⋯ 0 0 A 2 − 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n − 1 ] A^{-1}= \begin{bmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & A_n \end{bmatrix}^{-1} = \begin{bmatrix} A_1^{-1} & 0 & \cdots & 0 \\ 0 & A_2^{-1} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & A_n^{-1} \end{bmatrix} A−1= A10⋮00A2⋮0⋯⋯⋱⋯00⋮An −1= A1−10⋮00A2−1⋮0⋯⋯⋱⋯00⋮An−1
注： A i A_i Ai可逆，其中 i ∈ ( 1 , 2 , ⋯ , n ) i \in (1,2,\cdots,n) i∈(1,2,⋯,n)
4.1.4 反对角矩阵
A − 1 = [ 0 ⋯ 0 A 1 0 ⋯ A 2 0 ⋮ ⋱ ⋮ ⋮ A n ⋯ 0 0 ] − 1 = [ 0 ⋯ 0 A n − 1 0 ⋯ A n − 1 − 1 0 ⋮ ⋱ ⋮ ⋮ A 1 − 1 ⋯ 0 0 ] A^{-1}= \begin{bmatrix} 0 & \cdots & 0 & A_1 \\ 0 & \cdots & A_2 & 0 \\ \vdots & \ddots & \vdots & \vdots\\ A_n& \cdots & 0 & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & \cdots & 0 & A_n^{-1} \\ 0 & \cdots & A_{n-1}^{-1} & 0 \\ \vdots & \ddots & \vdots & \vdots\\ A_1^{-1}& \cdots & 0 & 0 \end{bmatrix} A−1= 00⋮An⋯⋯⋱⋯0A2⋮0A10⋮0 −1= 00⋮A1−1⋯⋯⋱⋯0An−1−1⋮0An−10⋮0
注： A i A_i Ai可逆，其中 i ∈ ( 1 , 2 , ⋯ , n ) i \in (1,2,\cdots,n) i∈(1,2,⋯,n)
4.2 反对称矩阵
4.2.1 定义
ω = [ ω x , ω y , ω z ] T \omega = [\omega_x, \omega_y, \omega_z]^T ω=[ωx,ωy,ωz]T，其反对称矩阵如下：
[ ω × ] = ω ∧ = A ω = [ 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 ] \left[ \omega_{\times} \right] = \omega^{\wedge} = A_{\omega} = \begin {bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end{bmatrix} [ω×]=ω∧=Aω= 0ωz−ωy−ωz0ωxωy−ωx0
ω = [ 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 ] ∨ \omega = \begin {bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end{bmatrix}^{\vee} ω= 0ωz−ωy−ωz0ωxωy−ωx0 ∨
4.2.2 交换子特性——李括号
反对称矩阵的交换子特性
令 v = [ v x , v y , v z ] T v = [v_x, v_y, v_z]^T v=[vx,vy,vz]T、 s = [ s x , s y , s z ] T s = [s_x, s_y, s_z]^T s=[sx,sy,sz]T，则
[ v × ] [ s × ] − [ s × ] [ v × ] = [ ( v × s ) × ] \left[ v_{\times} \right] \left[ s_{\times} \right] - \left[ s_{\times} \right] \left[ v_{\times} \right] = \left[ \left( v\times s \right)_{\times} \right] [v×][s×]−[s×][v×]=[(v×s)×]
4.2.3 二重积特性
令 s = [ s x , s y , s z ] T s = [s_x, s_y, s_z]^T s=[sx,sy,sz]T、 A s = s ∧ A_s = s^{\wedge} As=s∧、 ∥ s ∥ = s x 2 + s y 2 + s z 2 \lVert s \rVert = \sqrt{s_x^2 + s_y^2 + s_z^2} ∥s∥=sx2+sy2+sz2，
A s A s = s s T − s T s I 3 = s s T − ∥ s ∥ 2 I 3 A_s A_s = s s^T - s^T s I_3 = s s^T - \lVert s \rVert^2 I_3 AsAs=ssT−sTsI3=ssT−∥s∥2I3
4.2.4 三重积特性
令 s = [ s x , s y , s z ] T s = [s_x, s_y, s_z]^T s=[sx,sy,sz]T、 A s = s ∧ A_s = s^{\wedge} As=s∧、 ∥ s ∥ = s x 2 + s y 2 + s z 2 \lVert s \rVert = \sqrt{s_x^2 + s_y^2 + s_z^2} ∥s∥=sx2+sy2+sz2，则
A s A s A s = − ∥ s ∥ 2 A s A_s A_s A_s = -\lVert s \rVert ^2 A_s AsAsAs=−∥s∥2As
4.2.5 乘积迹
令 v = [ v x , v y , v z ] T v = [v_x, v_y, v_z]^T v=[vx,vy,vz]T、 s = [ s x , s y , s z ] T s = [s_x, s_y, s_z]^T s=[sx,sy,sz]T，则
t r ( [ v × ] [ s × ] ) = − 2 v ⋅ s tr\left( \left[ v_{\times} \right] \left[ s_{\times} \right] \right) = -2 v \cdot s tr([v×][s×])=−2v⋅s
5 群
常见群
复平面上的单位圆群：$ S 1 = { z ∈ C ∣ ∣ z ∣ = 1 } S^1 = \lbrace z \in \mathbb C | \lvert z \rvert=1 \rbrace S1={ z∈C∣∣z∣=1}
F \mathbb{F} F数域上的方阵群： M n ( F ) = { m ∈ F n } M_n(\mathbb F) = \lbrace m \in \mathbb F^n \rbrace Mn(F)={ m∈Fn}
一般线性群（general linear group）：$ G L n ( F ) = { m ∈ F n ∣ det ( m ) ≠ 0 } GL_n(\mathbb F) = \lbrace m \in \mathbb F^n | \det(m) \ne 0 \rbrace GLn(F)={ m∈Fn∣det(m)=0}
特殊线性群（special linear group）： S L n ( F ) = { m ∈ F n ∣ det ( m ) = 1 } SL_n(\mathbb F) = \lbrace m \in \mathbb F^n | \det(m) = 1 \rbrace SLn(F)={ m∈Fn∣det(m)=1}
正交群（orthogonal group）：$ O n ( F ) = { m ∈ F n ∣ m T m = I n } O_n(\mathbb F) = \lbrace m \in \mathbb F^n | m^T m = I_n \rbrace On(F)={ m∈Fn∣mTm=In}
特殊正交群（special orthogonal group）： S O n ( F ) = { m ∈ F n ∣ m T m = I n , det ( m ) = 1 } SO_n(\mathbb F) = \lbrace m \in \mathbb F^n | m^T m = I_n, \det(m) = 1 \rbrace SOn(F)={ m∈Fn∣mTm=In,det(m)=1}
酉群（unitary group）： U n ( F ) = { m ∈ F n ∣ m + m = I n } U_n(\mathbb F) = \lbrace m \in \mathbb F^n | m^+ m = I_n \rbrace Un(F)={ m∈Fn∣m+m=In}
特殊酉群（special unitary group）： S U n ( F ) = { m ∈ F n ∣ m + m = I n , det ( m ) = 1 } SU_n(\mathbb F) = \lbrace m \in \mathbb F^n | m^+ m = I_n, \det(m) = 1 \rbrace SUn(F)={ m∈Fn∣m+m=In,det(m)=1}
今天的文章 常见数学公式分享到此就结束了，感谢您的阅读。
版权声明：本文内容由互联网用户自发贡献，该文观点仅代表作者本人。本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如发现本站有涉嫌侵权/违法违规的内容， 请发送邮件至 举报，一经查实，本站将立刻删除。
如需转载请保留出处：https://bianchenghao.cn/bian-cheng-ji-chu/87172.html