深度学习常用数学知识

深度学习常用数学知识


为什么图片能被计算机读取?为什么我们可以用CNN对成千上万中图片进行分类,这背后的原理是什么?在了解原理之前,先给大家补点数学知识。因为无论是深度学习还是机器学习,背后都是有一些数学原理和公式推导的,所以掌握必备的数学知识必不可少,下面会给大家简单科普下常用的数学知识有哪些~

数学基础知识

数据科学需要一定的数学基础,但仅仅做应用的话,如果时间不多,不用学太深,了解基本公式即可,遇到问题再查吧。

下面是常见的一些数学基础概念,建议大家收藏后再仔细阅读,遇到不懂的概念可以直接在这里查~

高等数学

1.导数定义:

导数和微分的概念

f ′ ( x 0 ) = lim ⁡ Δ x → 0   f ( x 0 + Δ x ) − f ( x 0 ) Δ x f'({{x}_{0}})=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x} f(x0)=Δx0limΔxf(x0+Δx)f(x0) (1)

或者:

f ′ ( x 0 ) = lim ⁡ x → x 0   f ( x ) − f ( x 0 ) x − x 0 f'({{x}_{0}})=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}} f(x0)=xx0limxx0f(x)f(x0) (2)

2.左右导数导数的几何意义和物理意义

函数 f ( x ) f(x) f(x) x 0 x_0 x0处的左、右导数分别定义为:

左导数: f ′ − ( x 0 ) = lim ⁡ Δ x → 0 −   f ( x 0 + Δ x ) − f ( x 0 ) Δ x = lim ⁡ x → x 0 −   f ( x ) − f ( x 0 ) x − x 0 , ( x = x 0 + Δ x ) {{{f}'}_{-}}({{x}_{0}})=\underset{\Delta x\to {{0}^{-}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{-}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}},(x={{x}_{0}}+\Delta x) f(x0)=Δx0limΔxf(x0+Δx)f(x0)=xx0limxx0f(x)f(x0),(x=x0+Δx)

右导数: f ′ + ( x 0 ) = lim ⁡ Δ x → 0 +   f ( x 0 + Δ x ) − f ( x 0 ) Δ x = lim ⁡ x → x 0 +   f ( x ) − f ( x 0 ) x − x 0 {{{f}'}_{+}}({{x}_{0}})=\underset{\Delta x\to {{0}^{+}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{+}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}} f+(x0)=Δx0+limΔxf(x0+Δx)f(x0)=xx0+limxx0f(x)f(x0)

3.函数的可导性与连续性之间的关系

Th1: 函数 f ( x ) f(x) f(x) x 0 x_0 x0处可微 ⇔ f ( x ) \Leftrightarrow f(x) f(x) x 0 x_0 x0处可导

Th2: 若函数在点 x 0 x_0 x0处可导,则 y = f ( x ) y=f(x) y=f(x)在点 x 0 x_0 x0处连续,反之则不成立。即函数连续不一定可导。

Th3: f ′ ( x 0 ) {f}'({{x}_{0}}) f(x0)存在 ⇔ f ′ − ( x 0 ) = f ′ + ( x 0 ) \Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}}) f(x0)=f+(x0)

4.平面曲线的切线和法线

切线方程 : y − y 0 = f ′ ( x 0 ) ( x − x 0 ) y-{{y}_{0}}=f'({{x}_{0}})(x-{{x}_{0}}) yy0=f(x0)(xx0)
法线方程: y − y 0 = − 1 f ′ ( x 0 ) ( x − x 0 ) , f ′ ( x 0 ) ≠ 0 y-{{y}_{0}}=-\frac{1}{f'({{x}_{0}})}(x-{{x}_{0}}),f'({{x}_{0}})\ne 0 yy0=f(x0)1(xx0),f(x0)=0

5.四则运算法则
设函数 u = u ( x ) , v = v ( x ) u=u(x),v=v(x) u=u(x)v=v(x)]在点 x x x可导则
(1) ( u ± v ) ′ = u ′ ± v ′ (u\pm v{)}'={u}'\pm {v}' (u±v)=u±v d ( u ± v ) = d u ± d v d(u\pm v)=du\pm dv d(u±v)=du±dv
(2) ( u v ) ′ = u v ′ + v u ′ (uv{)}'=u{v}'+v{u}' (uv)=uv+vu d ( u v ) = u d v + v d u d(uv)=udv+vdu d(uv)=udv+vdu
(3) ( u v ) ′ = v u ′ − u v ′ v 2 ( v ≠ 0 ) (\frac{u}{v}{)}'=\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\ne 0) (vu)=v2vuuv(v=0) d ( u v ) = v d u − u d v v 2 d(\frac{u}{v})=\frac{vdu-udv}{{{v}^{2}}} d(vu)=v2vduudv

6.基本导数与微分表

(1) y = c y=c y=c(常数) y ′ = 0 {y}'=0 y=0 d y = 0 dy=0 dy=0

(2) y = x α y={{x}^{\alpha }} y=xα( α \alpha α为实数) y ′ = α x α − 1 {y}'=\alpha {{x}^{\alpha -1}} y=αxα1 d y = α x α − 1 d x dy=\alpha {{x}^{\alpha -1}}dx dy=αxα1dx

(3) y = a x y={{a}^{x}} y=ax y ′ = a x ln ⁡ a {y}'={{a}^{x}}\ln a y=axlna d y = a x ln ⁡ a d x dy={{a}^{x}}\ln adx dy=axlnadx
特例: ( e x ) ′ = e x ({{{e}}^{x}}{)}'={{{e}}^{x}} (ex)=ex d ( e x ) = e x d x d({{{e}}^{x}})={{{e}}^{x}}dx d(ex)=exdx

(4) y = log ⁡ a x y={{\log }_{a}}x y=logax y ′ = 1 x ln ⁡ a {y}'=\frac{1}{x\ln a} y=xlna1

d y = 1 x ln ⁡ a d x dy=\frac{1}{x\ln a}dx dy=xlna1dx
特例: y = ln ⁡ x y=\ln x y=lnx ( ln ⁡ x ) ′ = 1 x (\ln x{)}'=\frac{1}{x} (lnx)=x1 d ( ln ⁡ x ) = 1 x d x d(\ln x)=\frac{1}{x}dx d(lnx)=x1dx

(5) y = sin ⁡ x y=\sin x y=sinx

y ′ = cos ⁡ x {y}'=\cos x y=cosx d ( sin ⁡ x ) = cos ⁡ x d x d(\sin x)=\cos xdx d(sinx)=cosxdx

(6) y = cos ⁡ x y=\cos x y=cosx

y ′ = − sin ⁡ x {y}'=-\sin x y=sinx d ( cos ⁡ x ) = − sin ⁡ x d x d(\cos x)=-\sin xdx d(cosx)=sinxdx

(7) y = tan ⁡ x y=\tan x y=tanx

y ′ = 1 cos ⁡ 2 x = sec ⁡ 2 x {y}'=\frac{1}{{{\cos }^{2}}x}={{\sec }^{2}}x y=cos2x1=sec2x d ( tan ⁡ x ) = sec ⁡ 2 x d x d(\tan x)={{\sec }^{2}}xdx d(tanx)=sec2xdx
(8) y = cot ⁡ x y=\cot x y=cotx y ′ = − 1 sin ⁡ 2 x = − csc ⁡ 2 x {y}'=-\frac{1}{{{\sin }^{2}}x}=-{{\csc }^{2}}x y=sin2x1=csc2x d ( cot ⁡ x ) = − csc ⁡ 2 x d x d(\cot x)=-{{\csc }^{2}}xdx d(cotx)=csc2xdx
(9) y = sec ⁡ x y=\sec x y=secx y ′ = sec ⁡ x tan ⁡ x {y}'=\sec x\tan x y=secxtanx

d ( sec ⁡ x ) = sec ⁡ x tan ⁡ x d x d(\sec x)=\sec x\tan xdx d(secx)=secxtanxdx
(10) y = csc ⁡ x y=\csc x y=cscx y ′ = − csc ⁡ x cot ⁡ x {y}'=-\csc x\cot x y=cscxcotx

d ( csc ⁡ x ) = − csc ⁡ x cot ⁡ x d x d(\csc x)=-\csc x\cot xdx d(cscx)=cscxcotxdx
(11) y = arcsin ⁡ x y=\arcsin x y=arcsinx

y ′ = 1 1 − x 2 {y}'=\frac{1}{\sqrt{1-{{x}^{2}}}} y=1x2 1

d ( arcsin ⁡ x ) = 1 1 − x 2 d x d(\arcsin x)=\frac{1}{\sqrt{1-{{x}^{2}}}}dx d(arcsinx)=1x2 1dx
(12) y = arccos ⁡ x y=\arccos x y=arccosx

y ′ = − 1 1 − x 2 {y}'=-\frac{1}{\sqrt{1-{{x}^{2}}}} y=1x2 1 d ( arccos ⁡ x ) = − 1 1 − x 2 d x d(\arccos x)=-\frac{1}{\sqrt{1-{{x}^{2}}}}dx d(arccosx)=1x2 1dx

(13) y = arctan ⁡ x y=\arctan x y=arctanx

y ′ = 1 1 + x 2 {y}'=\frac{1}{1+{{x}^{2}}} y=1+x21 d ( arctan ⁡ x ) = 1 1 + x 2 d x d(\arctan x)=\frac{1}{1+{{x}^{2}}}dx d(arctanx)=1+x21dx

(14) y = arc ⁡ cot ⁡ x y=\operatorname{arc}\cot x y=arccotx

y ′ = − 1 1 + x 2 {y}'=-\frac{1}{1+{{x}^{2}}} y=1+x21

d ( arc ⁡ cot ⁡ x ) = − 1 1 + x 2 d x d(\operatorname{arc}\cot x)=-\frac{1}{1+{{x}^{2}}}dx d(arccotx)=1+x21dx
(15) y = s h x y=shx y=shx

y ′ = c h x {y}'=chx y=chx d ( s h x ) = c h x d x d(shx)=chxdx d(shx)=chxdx

(16) y = c h x y=chx y=chx

y ′ = s h x {y}'=shx y=shx d ( c h x ) = s h x d x d(chx)=shxdx d(chx)=shxdx

7.复合函数,反函数,隐函数以及参数方程所确定的函数的微分法

(1) 反函数的运算法则: 设 y = f ( x ) y=f(x) y=f(x)在点 x x x的某邻域内单调连续,在点 x x x处可导且 f ′ ( x ) ≠ 0 {f}'(x)\ne 0 f(x)=0,则其反函数在点 x x x所对应的 y y y处可导,并且有 d y d x = 1 d x d y \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}} dxdy=dydx1

(2) 复合函数的运算法则:若 μ = φ ( x ) \mu =\varphi(x) μ=φ(x) 在点 x x x可导,而 y = f ( μ ) y=f(\mu) y=f(μ)在对应点 μ \mu μ( μ = φ ( x ) \mu =\varphi (x) μ=φ(x))可导,则复合函数 y = f ( φ ( x ) ) y=f(\varphi (x)) y=f(φ(x))在点 x x x可导,且 y ′ = f ′ ( μ ) ⋅ φ ′ ( x ) {y}'={f}'(\mu )\cdot {\varphi }'(x) y=f(μ)φ(x)

(3) 隐函数导数 d y d x \frac{dy}{dx} dxdy的求法一般有三种方法:

1)方程两边对 x x x求导,要记住 y y y x x x的函数,则 y y y的函数是 x x x的复合函数.例如 1 y \frac{1}{y} y1 y 2 {{y}^{2}} y2 l n y ln y lny e y {{{e}}^{y}} ey等均是 x x x的复合函数.
x x x求导应按复合函数连锁法则做.

2)公式法.由 F ( x , y ) = 0 F(x,y)=0 F(x,y)=0 d y d x = − F ′ x ( x , y ) F ′ y ( x , y ) \frac{dy}{dx}=-\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)} dxdy=Fy(x,y)Fx(x,y),其中, F ′ x ( x , y ) {{{F}'}_{x}}(x,y) Fx(x,y)
F ′ y ( x , y ) {{{F}'}_{y}}(x,y) Fy(x,y)分别表示 F ( x , y ) F(x,y) F(x,y) x x x y y y的偏导数

3)利用微分形式不变性

8.常用高阶导数公式

(1) ( a x )   ( n ) = a x ln ⁡ n a ( a > 0 ) ( e x )   ( n ) = e   x ({{a}^{x}}){{\,}^{(n)}}={{a}^{x}}{{\ln }^{n}}a\quad (a>{0})\quad \quad ({{{e}}^{x}}){{\,}^{(n)}}={e}{{\,}^{x}} (ax)(n)=axlnna(a>0)(ex)(n)=ex

(2) ( sin ⁡ k x )   ( n ) = k n sin ⁡ ( k x + n ⋅ π 2 ) (\sin kx{)}{{\,}^{(n)}}={{k}^{n}}\sin (kx+n\cdot \frac{\pi }{{2}}) (sinkx)(n)=knsin(kx+n2π)

(3) ( cos ⁡ k x )   ( n ) = k n cos ⁡ ( k x + n ⋅ π 2 ) (\cos kx{)}{{\,}^{(n)}}={{k}^{n}}\cos (kx+n\cdot \frac{\pi }{{2}}) (coskx)(n)=kncos(kx+n2π)

(4) ( x m )   ( n ) = m ( m − 1 ) ⋯ ( m − n + 1 ) x m − n ({{x}^{m}}){{\,}^{(n)}}=m(m-1)\cdots (m-n+1){{x}^{m-n}} (xm)(n)=m(m1)(mn+1)xmn

(5) ( ln ⁡ x )   ( n ) = ( − 1 ) ( n − 1 ) ( n − 1 ) ! x n (\ln x){{\,}^{(n)}}={{(-{1})}^{(n-{1})}}\frac{(n-{1})!}{{{x}^{n}}} (lnx)(n)=(1)(n1)xn(n1)!

(6)莱布尼兹公式:若 u ( x )   , v ( x ) u(x)\,,v(x) u(x),v(x) n n n阶可导,则
( u v ) ( n ) = ∑ i = 0 n c n i u ( i ) v ( n − i ) {{(uv)}^{(n)}}=\sum\limits_{i={0}}^{n}{c_{n}^{i}{{u}^{(i)}}{{v}^{(n-i)}}} (uv)(n)=i=0ncniu(i)v(ni),其中 u ( 0 ) = u {{u}^{({0})}}=u u(0)=u v ( 0 ) = v {{v}^{({0})}}=v v(0)=v

9.微分中值定理,泰勒公式

Th1:(费马定理)

若函数 f ( x ) f(x) f(x)满足条件:
(1)函数 f ( x ) f(x) f(x) x 0 {{x}_{0}} x0的某邻域内有定义,并且在此邻域内恒有
f ( x ) ≤ f ( x 0 ) f(x)\le f({{x}_{0}}) f(x)f(x0) f ( x ) ≥ f ( x 0 ) f(x)\ge f({{x}_{0}}) f(x)f(x0),

(2) f ( x ) f(x) f(x) x 0 {{x}_{0}} x0处可导,则有 f ′ ( x 0 ) = 0 {f}'({{x}_{0}})=0 f(x0)=0

Th2:(罗尔定理)

设函数 f ( x ) f(x) f(x)满足条件:
(1)在闭区间 [ a , b ] [a,b] [a,b]上连续;

(2)在 ( a , b ) (a,b) (a,b)内可导;

(3) f ( a ) = f ( b ) f(a)=f(b) f(a)=f(b)

则在 ( a , b ) (a,b) (a,b)内一存在个 ξ \xi ξ,使 f ′ ( ξ ) = 0 {f}'(\xi )=0 f(ξ)=0

Th3: (拉格朗日中值定理)

设函数 f ( x ) f(x) f(x)满足条件:
(1)在 [ a , b ] [a,b] [a,b]上连续;

(2)在 ( a , b ) (a,b) (a,b)内可导;

则在 ( a , b ) (a,b) (a,b)内一存在个 ξ \xi ξ,使 f ( b ) − f ( a ) b − a = f ′ ( ξ ) \frac{f(b)-f(a)}{b-a}={f}'(\xi ) baf(b)f(a)=f(ξ)

Th4: (柯西中值定理)

设函数 f ( x ) f(x) f(x) g ( x ) g(x) g(x)满足条件:
(1) 在 [ a , b ] [a,b] [a,b]上连续;

(2) 在 ( a , b ) (a,b) (a,b)内可导且 f ′ ( x ) {f}'(x) f(x) g ′ ( x ) {g}'(x) g(x)均存在,且 g ′ ( x ) ≠ 0 {g}'(x)\ne 0 g(x)=0

则在 ( a , b ) (a,b) (a,b)内存在一个 ξ \xi ξ,使 f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( ξ ) g ′ ( ξ ) \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{{f}'(\xi )}{{g}'(\xi )} g(b)g(a)f(b)f(a)=g(ξ)f(ξ)

10.洛必达法则
法则Ⅰ ( 0 0 \frac{0}{0} 00型)
设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x)满足条件:
lim ⁡ x → x 0   f ( x ) = 0 , lim ⁡ x → x 0   g ( x ) = 0 \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=0 xx0limf(x)=0,xx0limg(x)=0;

f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x) x 0 {{x}_{0}} x0的邻域内可导,(在 x 0 {{x}_{0}} x0处可除外)且 g ′ ( x ) ≠ 0 {g}'\left( x \right)\ne 0 g(x)=0;

lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x)存在(或 ∞ \infty )。

则:
lim ⁡ x → x 0   f ( x ) g ( x ) = lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x)=xx0limg(x)f(x)
法则 I ′ {{I}'} I ( 0 0 \frac{0}{0} 00型)设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x)满足条件:
lim ⁡ x → ∞   f ( x ) = 0 , lim ⁡ x → ∞   g ( x ) = 0 \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to \infty }{\mathop{\lim }}\,g\left( x \right)=0 xlimf(x)=0,xlimg(x)=0;

存在一个 X > 0 X>0 X>0,当 ∣ x ∣ > X \left| x \right|>X x>X时, f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x)可导,且 g ′ ( x ) ≠ 0 {g}'\left( x \right)\ne 0 g(x)=0; lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x)存在(或 ∞ \infty )。

lim ⁡ x → x 0   f ( x ) g ( x ) = lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x)=xx0limg(x)f(x)
法则Ⅱ( ∞ ∞ \frac{\infty }{\infty } 型) 设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x) 满足条件:
lim ⁡ x → x 0   f ( x ) = ∞ , lim ⁡ x → x 0   g ( x ) = ∞ \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=\infty,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=\infty xx0limf(x)=,xx0limg(x)=;
f ( x ) , g ( x ) f\left( x \right),g\left( x \right) f(x),g(x) x 0 {{x}_{0}} x0 的邻域内可导(在 x 0 {{x}_{0}} x0处可除外)且 g ′ ( x ) ≠ 0 {g}'\left( x \right)\ne 0 g(x)=0; lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x) 存在(或 ∞ \infty )。
lim ⁡ x → x 0   f ( x ) g ( x ) = lim ⁡ x → x 0   f ′ ( x ) g ′ ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} xx0limg(x)f(x)=xx0limg(x)f(x) 同理法则 I I ′ {I{I}'} II ( ∞ ∞ \frac{\infty }{\infty } 型)仿法则 I ′ {{I}'} I 可写出。

11.泰勒公式

设函数 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处的某邻域内具有 n + 1 n+1 n+1阶导数,则对该邻域内异于 x 0 {{x}_{0}} x0的任意点 x x x,在 x 0 {{x}_{0}} x0 x x x之间至少存在
一个 ξ \xi ξ,使得:
f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 ! f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ⋯ f(x)=f({{x}_{0}})+{f}'({{x}_{0}})(x-{{x}_{0}})+\frac{1}{2!}{f}''({{x}_{0}}){{(x-{{x}_{0}})}^{2}}+\cdots f(x)=f(x0)+f(x0)(xx0)+2!1f(x0)(xx0)2+
+ f ( n ) ( x 0 ) n ! ( x − x 0 ) n + R n ( x ) +\frac{{{f}^{(n)}}({{x}_{0}})}{n!}{{(x-{{x}_{0}})}^{n}}+{{R}_{n}}(x) +n!f(n)(x0)(xx0)n+Rn(x)
其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 {{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{(x-{{x}_{0}})}^{n+1}} Rn(x)=(n+1)!f(n+1)(ξ)(xx0)n+1称为 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处的 n n n阶泰勒余项。

x 0 = 0 {{x}_{0}}=0 x0=0,则 n n n阶泰勒公式
f ( x ) = f ( 0 ) + f ′ ( 0 ) x + 1 2 ! f ′ ′ ( 0 ) x 2 + ⋯ + f ( n ) ( 0 ) n ! x n + R n ( x ) f(x)=f(0)+{f}'(0)x+\frac{1}{2!}{f}''(0){{x}^{2}}+\cdots +\frac{{{f}^{(n)}}(0)}{n!}{{x}^{n}}+{{R}_{n}}(x) f(x)=f(0)+f(0)x+2!1f(0)x2++n!f(n)(0)xn+Rn(x)……(1)
其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! x n + 1 {{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{x}^{n+1}} Rn(x)=(n+1)!f(n+1)(ξ)xn+1 ξ \xi ξ在0与 x x x之间.(1)式称为麦克劳林公式

常用五种函数在 x 0 = 0 {{x}_{0}}=0 x0=0处的泰勒公式

(1) e x = 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + x n + 1 ( n + 1 ) ! e ξ {{{e}}^{x}}=1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+\frac{{{x}^{n+1}}}{(n+1)!}{{e}^{\xi }} ex=1+x+2!1x2++n!1xn+(n+1)!xn+1eξ

= 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + o ( x n ) =1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+o({{x}^{n}}) =1+x+2!1x2++n!1xn+o(xn)

(2) sin ⁡ x = x − 1 3 ! x 3 + ⋯ + x n n ! sin ⁡ n π 2 + x n + 1 ( n + 1 ) ! sin ⁡ ( ξ + n + 1 2 π ) \sin x=x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\sin (\xi +\frac{n+1}{2}\pi ) sinx=x3!1x3++n!xnsin2nπ+(n+1)!xn+1sin(ξ+2n+1π)

= x − 1 3 ! x 3 + ⋯ + x n n ! sin ⁡ n π 2 + o ( x n ) =x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+o({{x}^{n}}) =x3!1x3++n!xnsin2nπ+o(xn)

(3) cos ⁡ x = 1 − 1 2 ! x 2 + ⋯ + x n n ! cos ⁡ n π 2 + x n + 1 ( n + 1 ) ! cos ⁡ ( ξ + n + 1 2 π ) \cos x=1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\cos (\xi +\frac{n+1}{2}\pi ) cosx=12!1x2++n!xncos2nπ+(n+1)!xn+1cos(ξ+2n+1π)

= 1 − 1 2 ! x 2 + ⋯ + x n n ! cos ⁡ n π 2 + o ( x n ) =1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+o({{x}^{n}}) =12!1x2++n!xncos2nπ+o(xn)

(4) ln ⁡ ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ⋯ + ( − 1 ) n − 1 x n n + ( − 1 ) n x n + 1 ( n + 1 ) ( 1 + ξ ) n + 1 \ln (1+x)=x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+\frac{{{(-1)}^{n}}{{x}^{n+1}}}{(n+1){{(1+\xi )}^{n+1}}} ln(1+x)=x21x2+31x3+(1)n1nxn+(n+1)(1+ξ)n+1(1)nxn+1

= x − 1 2 x 2 + 1 3 x 3 − ⋯ + ( − 1 ) n − 1 x n n + o ( x n ) =x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+o({{x}^{n}}) =x21x2+31x3+(1)n1nxn+o(xn)

(5) ( 1 + x ) m = 1 + m x + m ( m − 1 ) 2 ! x 2 + ⋯ + m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n {{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}} (1+x)m=1+mx+2!m(m1)x2++n!m(m1)(mn+1)xn
+ m ( m − 1 ) ⋯ ( m − n + 1 ) ( n + 1 ) ! x n + 1 ( 1 + ξ ) m − n − 1 +\frac{m(m-1)\cdots (m-n+1)}{(n+1)!}{{x}^{n+1}}{{(1+\xi )}^{m-n-1}} +(n+1)!m(m1)(mn+1)xn+1(1+ξ)mn1

( 1 + x ) m = 1 + m x + m ( m − 1 ) 2 ! x 2 + ⋯ {{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots (1+x)m=1+mx+2!m(m1)x2+ + m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n + o ( x n ) +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}+o({{x}^{n}}) +n!m(m1)(mn+1)xn+o(xn)

12.函数单调性的判断

Th1: 设函数 f ( x ) f(x) f(x) ( a , b ) (a,b) (a,b)区间内可导,如果对 ∀ x ∈ ( a , b ) \forall x\in (a,b) x(a,b),都有 f   ′ ( x ) > 0 f\,'(x)>0 f(x)>0(或 f   ′ ( x ) < 0 f\,'(x)<0 f(x)<0),则函数 f ( x ) f(x) f(x) ( a , b ) (a,b) (a,b)内是单调增加的(或单调减少)

Th2: (取极值的必要条件)设函数 f ( x ) f(x) f(x) x 0 {{x}_{0}} x0处可导,且在 x 0 {{x}_{0}} x0处取极值,则 f   ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f(x0)=0

Th3: (取极值的第一充分条件)设函数 f ( x ) f(x) f(x) x 0 {{x}_{0}} x0的某一邻域内可微,且 f   ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f(x0)=0(或 f ( x ) f(x) f(x) x 0 {{x}_{0}} x0处连续,但 f   ′ ( x 0 ) f\,'({{x}_{0}}) f(x0)不存在。)
(1)若当 x x x经过 x 0 {{x}_{0}} x0时, f   ′ ( x ) f\,'(x) f(x)由“+”变“-”,则 f ( x 0 ) f({{x}_{0}}) f(x0)为极大值;
(2)若当 x x x经过 x 0 {{x}_{0}} x0时, f   ′ ( x ) f\,'(x) f(x)由“-”变“+”,则 f ( x 0 ) f({{x}_{0}}) f(x0)为极小值;
(3)若 f   ′ ( x ) f\,'(x) f(x)经过 x = x 0 x={{x}_{0}} x=x0的两侧不变号,则 f ( x 0 ) f({{x}_{0}}) f(x0)不是极值。

Th4: (取极值的第二充分条件)设 f ( x ) f(x) f(x)在点 x 0 {{x}_{0}} x0处有 f ′ ′ ( x ) ≠ 0 f''(x)\ne 0 f(x)=0,且 f   ′ ( x 0 ) = 0 f\,'({{x}_{0}})=0 f(x0)=0,则 当 f ′   ′ ( x 0 ) < 0 f'\,'({{x}_{0}})<0 f(x0)<0时, f ( x 0 ) f({{x}_{0}}) f(x0)为极大值;
f ′   ′ ( x 0 ) > 0 f'\,'({{x}_{0}})>0 f(x0)>0时, f ( x 0 ) f({{x}_{0}}) f(x0)为极小值。
注:如果 f ′   ′ ( x 0 ) = 0 f'\,'({{x}_{0}})=0 f(x0)=0,此方法失效。

13.渐近线的求法
(1)水平渐近线 若 lim ⁡ x → + ∞   f ( x ) = b \underset{x\to +\infty }{\mathop{\lim }}\,f(x)=b x+limf(x)=b,或 lim ⁡ x → − ∞   f ( x ) = b \underset{x\to -\infty }{\mathop{\lim }}\,f(x)=b xlimf(x)=b,则

y = b y=b y=b称为函数 y = f ( x ) y=f(x) y=f(x)的水平渐近线。

(2)铅直渐近线 若 lim ⁡ x → x 0 −   f ( x ) = ∞ \underset{x\to x_{0}^{-}}{\mathop{\lim }}\,f(x)=\infty xx0limf(x)=,或 lim ⁡ x → x 0 +   f ( x ) = ∞ \underset{x\to x_{0}^{+}}{\mathop{\lim }}\,f(x)=\infty xx0+limf(x)=,则

x = x 0 x={{x}_{0}} x=x0称为 y = f ( x ) y=f(x) y=f(x)的铅直渐近线。

(3)斜渐近线 若 a = lim ⁡ x → ∞   f ( x ) x , b = lim ⁡ x → ∞   [ f ( x ) − a x ] a=\underset{x\to \infty }{\mathop{\lim }}\,\frac{f(x)}{x},\quad b=\underset{x\to \infty }{\mathop{\lim }}\,[f(x)-ax] a=xlimxf(x),b=xlim[f(x)ax],则
y = a x + b y=ax+b y=ax+b称为 y = f ( x ) y=f(x) y=f(x)的斜渐近线。

14.函数凹凸性的判断
Th1: (凹凸性的判别定理)若在I上 f ′ ′ ( x ) < 0 f''(x)<0 f(x)<0(或 f ′ ′ ( x ) > 0 f''(x)>0 f(x)>0),则 f ( x ) f(x) f(x)在I上是凸的(或凹的)。

Th2: (拐点的判别定理1)若在 x 0 {{x}_{0}} x0 f ′ ′ ( x ) = 0 f''(x)=0 f(x)=0,(或 f ′ ′ ( x ) f''(x) f(x)不存在),当 x x x变动经过 x 0 {{x}_{0}} x0时, f ′ ′ ( x ) f''(x) f(x)变号,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) (x0,f(x0))为拐点。

Th3: (拐点的判别定理2)设 f ( x ) f(x) f(x) x 0 {{x}_{0}} x0点的某邻域内有三阶导数,且 f ′ ′ ( x ) = 0 f''(x)=0 f(x)=0 f ′ ′ ′ ( x ) ≠ 0 f'''(x)\ne 0 f(x)=0,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) (x0,f(x0))为拐点。

15.弧微分

d S = 1 + y ′ 2 d x dS=\sqrt{1+y{{'}^{2}}}dx dS=1+y2 dx

16.曲率

曲线 y = f ( x ) y=f(x) y=f(x)在点 ( x , y ) (x,y) (x,y)处的曲率 k = ∣ y ′ ′ ∣ ( 1 + y ′ 2 ) 3 2 k=\frac{\left| y'' \right|}{{{(1+y{{'}^{2}})}^{\tfrac{3}{2}}}} k=(1+y2)23y
对于参数方程 { x = φ ( t ) y = ψ ( t ) , \left\{\begin{matrix}x=\varphi(t) \\ y=\psi (t) \end{matrix}\right., {x=φ(t)y=ψ(t),
k = ∣ φ ′ ( t ) ψ ′ ′ ( t ) − φ ′ ′ ( t ) ψ ′ ( t ) ∣ [ φ ′ 2 ( t ) + ψ ′ 2 ( t ) ] 3 2 k=\frac{\left| \varphi '(t)\psi ''(t)-\varphi ''(t)\psi '(t) \right|}{{{[\varphi {{'}^{2}}(t)+\psi {{'}^{2}}(t)]}^{\tfrac{3}{2}}}} k=[φ2(t)+ψ2(t)]23φ(t)ψ(t)φ(t)ψ(t)

17.曲率半径

曲线在点 M M M处的曲率 k ( k ≠ 0 ) k(k\ne 0) k(k=0)与曲线在点 M M M处的曲率半径 ρ \rho ρ有如下关系: ρ = 1 k \rho =\frac{1}{k} ρ=k1

线性代数

行列式

1.行列式按行(列)展开定理

(1) 设 A = ( a i j ) n × n A = ( a_{{ij}} )_{n \times n} A=(aij)n×n,则: a i 1 A j 1 + a i 2 A j 2 + ⋯ + a i n A j n = { ∣ A ∣ , i = j 0 , i ≠ j a_{i1}A_{j1} +a_{i2}A_{j2} + \cdots + a_{{in}}A_{{jn}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases} ai1Aj1+ai2Aj2++ainAjn={A,i=j0,i=j

a 1 i A 1 j + a 2 i A 2 j + ⋯ + a n i A n j = { ∣ A ∣ , i = j 0 , i ≠ j a_{1i}A_{1j} + a_{2i}A_{2j} + \cdots + a_{{ni}}A_{{nj}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases} a1iA1j+a2iA2j++aniAnj={A,i=j0,i=j A A ∗ = A ∗ A = ∣ A ∣ E , AA^{*} = A^{*}A = \left| A \right|E, AA=AA=AE,其中: A ∗ = ( A 11 A 12 … A 1 n A 21 A 22 … A 2 n … … … … A n 1 A n 2 … A n n ) = ( A j i ) = ( A i j ) T A^{*} = \begin{pmatrix} A_{11} & A_{12} & \ldots & A_{1n} \\ A_{21} & A_{22} & \ldots & A_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ A_{n1} & A_{n2} & \ldots & A_{{nn}} \\ \end{pmatrix} = (A_{{ji}}) = {(A_{{ij}})}^{T} A=A11A21An1A12A22An2A1nA2nAnn=(Aji)=(Aij)T

D n = ∣ 1 1 … 1 x 1 x 2 … x n … … … … x 1 n − 1 x 2 n − 1 … x n n − 1 ∣ = ∏ 1 ≤ j < i ≤ n   ( x i − x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n - 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j}) Dn=1x1x1n11x2x2n11xnxnn1=1j<in(xixj)

(2) 设 A , B A,B A,B n n n阶方阵,则 ∣ A B ∣ = ∣ A ∣ ∣ B ∣ = ∣ B ∣ ∣ A ∣ = ∣ B A ∣ \left| {AB} \right| = \left| A \right|\left| B \right| = \left| B \right|\left| A \right| = \left| {BA} \right| AB=AB=BA=BA,但 ∣ A ± B ∣ = ∣ A ∣ ± ∣ B ∣ \left| A \pm B \right| = \left| A \right| \pm \left| B \right| A±B=A±B不一定成立。

(3) ∣ k A ∣ = k n ∣ A ∣ \left| {kA} \right| = k^{n}\left| A \right| kA=knA, A A A n n n阶方阵。

(4) 设 A A A n n n阶方阵, ∣ A T ∣ = ∣ A ∣ ; ∣ A − 1 ∣ = ∣ A ∣ − 1 |A^{T}| = |A|;|A^{- 1}| = |A|^{- 1} AT=A;A1=A1(若 A A A可逆), ∣ A ∗ ∣ = ∣ A ∣ n − 1 |A^{*}| = |A|^{n - 1} A=An1

n ≥ 2 n \geq 2 n2

(5) ∣ A O O B ∣ = ∣ A C O B ∣ = ∣ A O C B ∣ = ∣ A ∣ ∣ B ∣ \left| \begin{matrix} & {A\quad O} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad C} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad O} \\ & {C\quad B} \\ \end{matrix} \right| =| A||B| AOOB=ACOB=AOCB=AB
A , B A,B A,B为方阵,但 ∣ O A m × m B n × n O ∣ = ( − 1 ) m n ∣ A ∣ ∣ B ∣ \left| \begin{matrix} {O} & A_{m \times m} \\ B_{n \times n} & { O} \\ \end{matrix} \right| = ({- 1)}^{{mn}}|A||B| OBn×nAm×mO=(1)mnAB

(6) 范德蒙行列式 D n = ∣ 1 1 … 1 x 1 x 2 … x n … … … … x 1 n − 1 x 2 n 1 … x n n − 1 ∣ = ∏ 1 ≤ j < i ≤ n   ( x i − x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j}) Dn=1x1x1n11x2x2n11xnxnn1=1j<in(xixj)

A A A n n n阶方阵, λ i ( i = 1 , 2 ⋯   , n ) \lambda_{i}(i = 1,2\cdots,n) λi(i=1,2,n) A A A n n n个特征值,则
∣ A ∣ = ∏ i = 1 n λ i |A| = \prod_{i = 1}^{n}\lambda_{i} A=i=1nλi

矩阵

矩阵: m × n m \times n m×n个数 a i j a_{{ij}} aij排成 m m m n n n列的表格 [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ ⋯ a m 1 a m 2 ⋯ a m n ] \begin{bmatrix} a_{11}\quad a_{12}\quad\cdots\quad a_{1n} \\ a_{21}\quad a_{22}\quad\cdots\quad a_{2n} \\ \quad\cdots\cdots\cdots\cdots\cdots \\ a_{m1}\quad a_{m2}\quad\cdots\quad a_{{mn}} \\ \end{bmatrix} a11a12a1na21a22a2nam1am2amn 称为矩阵,简记为 A A A,或者 ( a i j ) m × n \left( a_{{ij}} \right)_{m \times n} (aij)m×n 。若 m = n m = n m=n,则称 A A A n n n阶矩阵或 n n n阶方阵。

矩阵的线性运算

1.矩阵的加法

A = ( a i j ) , B = ( b i j ) A = (a_{{ij}}),B = (b_{{ij}}) A=(aij),B=(bij)是两个 m × n m \times n m×n矩阵,则 m × n m \times n m×n 矩阵 C = c i j ) = a i j + b i j C = c_{{ij}}) = a_{{ij}} + b_{{ij}} C=cij)=aij+bij称为矩阵 A A A B B B的和,记为 A + B = C A + B = C A+B=C

2.矩阵的数乘

A = ( a i j ) A = (a_{{ij}}) A=(aij) m × n m \times n m×n矩阵, k k k是一个常数,则 m × n m \times n m×n矩阵 ( k a i j ) (ka_{{ij}}) (kaij)称为数 k k k与矩阵 A A A的数乘,记为 k A {kA} kA

3.矩阵的乘法

A = ( a i j ) A = (a_{{ij}}) A=(aij) m × n m \times n m×n矩阵, B = ( b i j ) B = (b_{{ij}}) B=(bij) n × s n \times s n×s矩阵,那么 m × s m \times s m×s矩阵 C = ( c i j ) C = (c_{{ij}}) C=(cij),其中 c i j = a i 1 b 1 j + a i 2 b 2 j + ⋯ + a i n b n j = ∑ k = 1 n a i k b k j c_{{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{{in}}b_{{nj}} = \sum_{k =1}^{n}{a_{{ik}}b_{{kj}}} cij=ai1b1j+ai2b2j++ainbnj=k=1naikbkj称为 A B {AB} AB的乘积,记为 C = A B C = AB C=AB

4. A T \mathbf{A}^{\mathbf{T}} AT A − 1 \mathbf{A}^{\mathbf{-1}} A1 A ∗ \mathbf{A}^{\mathbf{*}} A三者之间的关系

(1) ( A T ) T = A , ( A B ) T = B T A T , ( k A ) T = k A T , ( A ± B ) T = A T ± B T {(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \pm B)}^{T} = A^{T} \pm B^{T} (AT)T=A,(AB)T=BTAT,(kA)T=kAT,(A±B)T=AT±BT

(2) ( A − 1 ) − 1 = A , ( A B ) − 1 = B − 1 A − 1 , ( k A ) − 1 = 1 k A − 1 , \left( A^{- 1} \right)^{- 1} = A,\left( {AB} \right)^{- 1} = B^{- 1}A^{- 1},\left( {kA} \right)^{- 1} = \frac{1}{k}A^{- 1}, (A1)1=A,(AB)1=B1A1,(kA)1=k1A1,

( A ± B ) − 1 = A − 1 ± B − 1 {(A \pm B)}^{- 1} = A^{- 1} \pm B^{- 1} (A±B)1=A1±B1不一定成立。

(3) ( A ∗ ) ∗ = ∣ A ∣ n − 2   A    ( n ≥ 3 ) \left( A^{*} \right)^{*} = |A|^{n - 2}\ A\ \ (n \geq 3) (A)=An2 A  (n3) ( A B ) ∗ = B ∗ A ∗ , \left({AB} \right)^{*} = B^{*}A^{*}, (AB)=BA, ( k A ) ∗ = k n − 1 A ∗    ( n ≥ 2 ) \left( {kA} \right)^{*} = k^{n -1}A^{*}{\ \ }\left( n \geq 2 \right) (kA)=kn1A  (n2)

( A ± B ) ∗ = A ∗ ± B ∗ \left( A \pm B \right)^{*} = A^{*} \pm B^{*} (A±B)=A±B不一定成立。

(4) ( A − 1 ) T = ( A T ) − 1 ,   ( A − 1 ) ∗ = ( A A ∗ ) − 1 , ( A ∗ ) T = ( A T ) ∗ {(A^{- 1})}^{T} = {(A^{T})}^{- 1},\ \left( A^{- 1} \right)^{*} ={(AA^{*})}^{- 1},{(A^{*})}^{T} = \left( A^{T} \right)^{*} (A1)T=(AT)1, (A1)=(AA)1,(A)T=(AT)

5.有关 A ∗ \mathbf{A}^{\mathbf{*}} A的结论

(1) A A ∗ = A ∗ A = ∣ A ∣ E AA^{*} = A^{*}A = |A|E AA=AA=AE

(2) ∣ A ∗ ∣ = ∣ A ∣ n − 1   ( n ≥ 2 ) ,      ( k A ) ∗ = k n − 1 A ∗ ,    ( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 3 ) |A^{*}| = |A|^{n - 1}\ (n \geq 2),\ \ \ \ {(kA)}^{*} = k^{n -1}A^{*},{{\ \ }\left( A^{*} \right)}^{*} = |A|^{n - 2}A(n \geq 3) A=An1 (n2),    (kA)=kn1A,  (A)=An2A(n3)

(3) 若 A A A可逆,则 A ∗ = ∣ A ∣ A − 1 , ( A ∗ ) ∗ = 1 ∣ A ∣ A A^{*} = |A|A^{- 1},{(A^{*})}^{*} = \frac{1}{|A|}A A=AA1,(A)=A1A

(4) 若 A A A n n n阶方阵,则:

r ( A ∗ ) = { n , r ( A ) = n 1 , r ( A ) = n − 1 0 , r ( A ) < n − 1 r(A^*)=\begin{cases}n,\quad r(A)=n\\ 1,\quad r(A)=n-1\\ 0,\quad r(A)r(A)=n,r(A)=n1,r(A)=n10,r(A)<n1

6.有关 A − 1 \mathbf{A}^{\mathbf{- 1}} A1的结论

A A A可逆 ⇔ A B = E ; ⇔ ∣ A ∣ ≠ 0 ; ⇔ r ( A ) = n ; \Leftrightarrow AB = E; \Leftrightarrow |A| \neq 0; \Leftrightarrow r(A) = n; AB=E;A=0;r(A)=n;

⇔ A \Leftrightarrow A A可以表示为初等矩阵的乘积; ⇔ A x = 0 \Leftrightarrow Ax = 0 Ax=0只有零解。

7.有关矩阵秩的结论

(1) 秩 r ( A ) r(A) r(A)=行秩=列秩;

(2) r ( A m × n ) ≤ min ⁡ ( m , n ) ; r(A_{m \times n}) \leq \min(m,n); r(Am×n)min(m,n);

(3) A ≠ 0 ⇒ r ( A ) ≥ 1 A \neq 0 \Rightarrow r(A) \geq 1 A=0r(A)1

(4) r ( A ± B ) ≤ r ( A ) + r ( B ) ; r(A \pm B) \leq r(A) + r(B); r(A±B)r(A)+r(B);

(5) 初等变换不改变矩阵的秩

(6) r ( A ) + r ( B ) − n ≤ r ( A B ) ≤ min ⁡ ( r ( A ) , r ( B ) ) , r(A) + r(B) - n \leq r(AB) \leq \min(r(A),r(B)), r(A)+r(B)nr(AB)min(r(A),r(B)),特别若 A B = O AB = O AB=O
则: r ( A ) + r ( B ) ≤ n r(A) + r(B) \leq n r(A)+r(B)n

(7) 若 A − 1 A^{- 1} A1存在 ⇒ r ( A B ) =

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