Matlab基础

基本运算：

一,矩阵的生成

clc ;
clear all;
close all;

1.直接输入

A = [ 1 ,2 ,3,4;2,3,4,5;3,4,5,6]

A =

     1     2     3     4
     2     3     4     5
     3     4     5     6

1)矩阵元素可以是表达式

B = [sin(pi/2),4*8]

B =

     1    32

2)矩阵元素可以是复数

C = [3+2i,2+6i,4-3i]

C =

   3.0000 + 2.0000i   2.0000 + 6.0000i   4.0000 - 3.0000i

2.使用函数建立矩阵

1)使用reshpe函数

a=1:9;
b = reshape(a,3,3)
c = reshape(a,3,3)'  % c为b的转置.

b =

     1     4     7
     2     5     8
     3     6     9


c =

     1     2     3
     4     5     6
     7     8     9

2)使用diag建立对角矩阵

A = rand(4,4)
B = diag(A)  % 这里B并不是对角矩阵
C =diag(B)    % c才是A中元素的对角部分.

A =

    0.4299    0.3968    0.2160    0.6713
    0.8878    0.8085    0.7904    0.4386
    0.3912    0.7551    0.9493    0.8335
    0.7691    0.3774    0.3276    0.7689


B =

    0.4299
    0.8085
    0.9493
    0.7689


C =

    0.4299         0         0         0
         0    0.8085         0         0
         0         0    0.9493         0
         0         0         0    0.7689

3.使用M文件建立矩阵

二.算术运算

1.矩阵的加减运算.

1)加减运算

A=[1,2;3,4];
B=[5,6;7,8];
x1=A+B
x2=A-B

x1 =

     6     8
    10    12


x2 =

    -4    -4
    -4    -4

2)乘法

A=[2 5 7 ;1 3 6 ;3 2 5 ];
B=[1 3 5 ; 2 4 6 ;5 6 7 ];
c1=A*B  %矩阵相乘必须有A为[m,n],B为[n,p]矩阵,即为A的列数需要和B的行数相等.
c2=B*A

c1 =

    47    68    89
    37    51    65
    32    47    62


c2 =

    20    24    50
    26    34    68
    37    57   106

3)除法

%矩阵的除法分为左除和右除,
% AB=inv(A)*B
% B/A = B*inv(A)
a=[11 5 3; 78 5 21; 7 15 9 ];
b=[20 30 40;30 40 50 ; 40 50 60];
c1=a
c2=b/a
d1=inv(a)*b
d2=b*inv(a)
%比较c1,d1和c2,d2,既可以看到是否相等.

c1 =

    0.7692    1.5385    2.3077
    3.6923    5.7179    7.7436
   -2.3077   -5.1709   -8.0342


c2 =

   -9.9786    1.2222    4.9188
  -11.3034    1.4444    5.9530
  -12.6282    1.6667    6.9872


d1 =

    0.7692    1.5385    2.3077
    3.6923    5.7179    7.7436
   -2.3077   -5.1709   -8.0342


d2 =

   -9.9786    1.2222    4.9188
  -11.3034    1.4444    5.9530
  -12.6282    1.6667    6.9872

4)矩阵的乘方

%乘方的要求为:A^x,其中A为方阵,x为标量.
A=[3 0 7; 9 12 8; 1 5 3];
result = A^3    %A为方阵,x为标量.

result =

         405         630         518
        2268        3123        3022
         884        1180        1125

5)点运算

% 点运算分为 .* 、 ./ 、.、 .^
%【两个矩阵进行点运算】是指对应元素进行运算。
%A.*B 指的是对应元素相乘.
%A./B 表示对应元素相除
%A.^B表示对应元素做乘方运算.
%【要求】两矩阵具有相同的维数。

A = [1 3 5;2 4 6; 3 6 9]
B= [9 10 7 ; 6 9 4; 2 5 8]
C=A.*B            % A.*B 指的是对应元素相乘.

% A./B == B.A
C1=A./B           %A./B 表示对应元素相除
C2=B.A

x=[2 2 2 ]
y=[1 2 3 ]
z=x.^y            %x.^y表示对应元素做乘方运算.

A =

     1     3     5
     2     4     6
     3     6     9


B =

     9    10     7
     6     9     4
     2     5     8


C =

     9    30    35
    12    36    24
     6    30    72


C1 =

    0.1111    0.3000    0.7143
    0.3333    0.4444    1.5000
    1.5000    1.2000    1.1250


C2 =

    0.1111    0.3000    0.7143
    0.3333    0.4444    1.5000
    1.5000    1.2000    1.1250


x =

     2     2     2


y =

     1     2     3


z =

     2     4     8

【例1-3】当x=1,3,5,7,9时,分别求y=x^2cos(x)的值.

x=1:2:9
y=x.^2.*cos(x)

x =

     1     3     5     7     9


y =

    0.5403   -8.9099    7.0916   36.9412  -73.8016

三.关系运算

关系运算符有 == ~= < <= > >=

% 关系运算是以矩阵内元素对元素的方式做运算的.
a=[1 2 ; 3 4]
b=[1 3;2 4]
c1 = a>b  %矩阵对应元素比较
c2 = a==b
c3 = a~=b
c4 = a<b
c5 = a<=b
c6 = a>=b
d= a(c6)  %取出a中非零的元素.

a =

     1     2
     3     4


b =

     1     3
     2     4


c1 =

     0     0
     1     0


c2 =

     1     0
     0     1


c3 =

     0     1
     1     0


c4 =

     0     1
     0     0


c5 =

     1     1
     0     1


c6 =

     1     0
     1     1


d =

     1
     3
     4

四.逻辑运算

% 逻辑运算有与(&),或(|),非(~)
% and(a,b) or(a,b) not(a) xor(a,b)
a = [17 26 39 3 -32]
b = [23 168 58 -3 25]
c1 = a>10
c2 = b<10
d1 = a>10 & b<10
% 追个比较: a>10的结果是 [1 ,1 ,1,0,0]
%           b<10的结果是[0 ,0 , 0,1,0]
%上面两个结果相与操作是  [0 , 0, 0,0,0]

% 关系与逻辑运算函数

a =

    17    26    39     3   -32


b =

    23   168    58    -3    25


c1 =

     1     1     1     0     0


c2 =

     0     0     0     1     0


d1 =

     0     0     0     0     0

1.all 若向量的所有元素非零,结果为1

resultALL = all([10 20 30 40 ]) %均非零,结果为1.

resultALL =

     1

2.any 若向量中任何一个元素非零,结果为1

resultANY = any([1 2 3 4 5 0]) %有元素非零,结果为1.

resultANY =

     1

3.exist 判断工作空间是否存在,存在为1,否则非零.

resultEXIST = exist('plot')

resultEXIST =

     5

4.find 找出向量或矩阵非零元素的位置

dataforFIND = [2 1 0; 6 0 3 ;5 2 4 ]
resultFIND1 = find(dataforFIND)
resultFIND2 = find(dataforFIND')
% 从结果来看,find的顺序是
order = [ 1 4 7;2 5 8 ; 3 6 9]

dataforFIND =

     2     1     0
     6     0     3
     5     2     4


resultFIND1 =

     1
     2
     3
     4
     6
     8
     9


resultFIND2 =

     1
     2
     4
     6
     7
     8
     9


order =

     1     4     7
     2     5     8
     3     6     9

5.isempty 判断矩阵是空矩阵 如果为空,则为1,非空0;

resultISEMPTY = isempty([1 2 3 ; 4 5 6 ; 7 8 9 ]) %矩阵显然非空,则为0

resultISEMPTY =

     0

6.isglobal 判断是否为全局变量,是为1,非为0

7.isinf 判断是否为无穷小量,是为1,否为0

8.isnan 若元素是nan,则为1,否为0

9.isfinite 若矩阵大小有限,则取1,否则则为0

10. issparse 判断是否为稀疏矩阵,是为1,否为0

11. isstr 判断变量是字符组,是为1,否则取0

五.基本函数

1.常用数学函数

1)绝对值函数abs(a)

x = [0 -1 -2 4 -4 9]
absOFx = abs(x)

x =

     0    -1    -2     4    -4     9


absOFx =

     0     1     2     4     4     9

2)相位角函数angle(a) 以弧度表示.

a=[3+4i,5+5i;5+12i 3]
angleOFa=angle(a)*(180/pi)  %将弧度转换为度数.

a =

   3.0000 + 4.0000i   5.0000 + 5.0000i
   5.0000 +12.0000i   3.0000          


angleOFa =

   53.1301   45.0000
   67.3801         0

3)求复数的实部和虚部real(a),imag(a)

a=[7-8i 10+i;3 7+2i;12-6i 3]
realOFa = real(a)
imagOFa = imag(a)
realANDimag = real(a) + imag(a)*i

a =

   7.0000 - 8.0000i  10.0000 + 1.0000i
   3.0000             7.0000 + 2.0000i
  12.0000 - 6.0000i   3.0000          


realOFa =

     7    10
     3     7
    12     3


imagOFa =

    -8     1
     0     2
    -6     0


realANDimag =

   7.0000 - 8.0000i  10.0000 + 1.0000i
   3.0000             7.0000 + 2.0000i
  12.0000 - 6.0000i   3.0000          

4)求复数共轭conj(a)

a=[7-8i 10+i;3 7+2i;12-6i 3]
conjOFa = conj(a)
realANDimag = real(a) - imag(a)*j

a =

   7.0000 - 8.0000i  10.0000 + 1.0000i
   3.0000             7.0000 + 2.0000i
  12.0000 - 6.0000i   3.0000          


conjOFa =

   7.0000 + 8.0000i  10.0000 - 1.0000i
   3.0000             7.0000 - 2.0000i
  12.0000 + 6.0000i   3.0000          


realANDimag =

   7.0000 + 8.0000i  10.0000 - 1.0000i
   3.0000             7.0000 - 2.0000i
  12.0000 + 6.0000i   3.0000          

5)数组指数exp(a) 将矩阵元素作为e的幂exp(a)=e.^a

a = [1 -3 3 ;2 -1 6]
resultOFa = exp(a)

a =

     1    -3     3
     2    -1     6


resultOFa =

    2.7183    0.0498   20.0855
    7.3891    0.3679  403.4288

6)矩阵指数函数的使用expm(a)=e^a

a = [0 2 0 ;-2 0 3 ; 0 2 -1]
resultOFexpm = expm(a)
resultOFexp = exp(1)^a

a =

     0     2     0
    -2     0     3
     0     2    -1


resultOFexpm =

   -1.1709    2.2727    2.4144
   -2.2727    1.2435    2.2018
   -1.6096    1.4679    2.1191


resultOFexp =

  -1.1709             2.2727 + 0.0000i   2.4144 + 0.0000i
  -2.2727 - 0.0000i   1.2435 + 0.0000i   2.2018 + 0.0000i
  -1.6096 + 0.0000i   1.4679 + 0.0000i   2.1191 + 0.0000i

7)平方根函数的使用 sqrt(a)

a=[2 5 8 9]
resultOFsqrt = sqrt(a)

a =

     2     5     8     9


resultOFsqrt =

    1.4142    2.2361    2.8284    3.0000

8)对数函数 自然对数log(a),log2(a),log10(a)

a = [12 3 7;-1 4 -2; 21 17 -5]
resultOFlog = log(a)
resultOFlog2 = log2(a)
resultOFlog10 = log10(a)

a =

    12     3     7
    -1     4    -2
    21    17    -5


resultOFlog =

   2.4849             1.0986             1.9459          
        0 + 3.1416i   1.3863             0.6931 + 3.1416i
   3.0445             2.8332             1.6094 + 3.1416i


resultOFlog2 =

   3.5850             1.5850             2.8074          
        0 + 4.5324i   2.0000             1.0000 + 4.5324i
   4.3923             4.0875             2.3219 + 4.5324i


resultOFlog10 =

   1.0792             0.4771             0.8451          
        0 + 1.3644i   0.6021             0.3010 + 1.3644i
   1.3222             1.2304             0.6990 + 1.3644i

9)舎入函数 round(a),floor(a),ceil(a),fix(a)

a = [0.5 0.1 -0.2; -0.8 0.3 1.6]
% round(a)  | ←!→|   向最接近的整数舍入.
resultOFround = round(a)

% floor(a)  |←!|     向负无穷方向舍入.
resultOFfloor = floor(a)

% ceil(a)   |!→|     向正无穷方向舍入.
resultOFceil = ceil(a)

% fix(a)    !→|0|←! 向零方向舍入.
resultOFfix = fix(a)

a =

    0.5000    0.1000   -0.2000
   -0.8000    0.3000    1.6000


resultOFround =

     1     0     0
    -1     0     2


resultOFfloor =

     0     0    -1
    -1     0     1


resultOFceil =

     1     1     0
     0     1     2


resultOFfix =

     0     0     0
     0     0     1

10)模除求余数函数的使用mod(x,y),rem(x,y)

% mod(x,y) 表示x对y取模, mod(x,y) = x-y.*floor(x./y)
% rem(x,y) 表示x对y求余数,rem(x,y) = x-y.*fix(x./y)
a = [5 6 7;-3 -2 -1;18 -16 9]
b = [2 3 -2; 1 5 6;-3 -1 7]
ResultOfAMod3 = mod(a,3)
ResultOfAModB = mod(a,b)
ResultOfARemB = rem(a,b)

a =

     5     6     7
    -3    -2    -1
    18   -16     9


b =

     2     3    -2
     1     5     6
    -3    -1     7


ResultOfAMod3 =

     2     0     1
     0     1     2
     0     2     0


ResultOfAModB =

     1     0    -1
     0     3     5
     0     0     2


ResultOfARemB =

     1     0     1
     0    -2    -1
     0     0     2

2.随机函数的使用

1)均匀分布随机矩阵函数

m = 4
n = 5
x = rand(n) %生成n*n随机矩阵,其元素在0-1内.
x = rand(m,n) %生成m*n随机矩阵.
x = rand([m,n]) %生成m*n随机矩阵
% x = rand(m,n,p...) %生成m*n*p*...随机矩阵
% x = rand([m n p ... ]) %生成m*n*p*...随机矩阵
x = rand(size(A)) %生成与矩阵A相同大小的随机矩阵.
resultofRAND1 = rand  %无变量输入时只产生一个随机数
resultofRAND2 = rand('state')  %产生包括均衡发生器当前状态的35个随机数
% resultofRAND3 = rand('state',s) %状态重置为s
% resultofRAND4 = rand('state',0) %重置发生器到初始状态
% resultofRAND5 = rand('state',j) %对整数j重置发生器到第j个状态
% resultofRAND6 = rand('state',sum(100*clock)) % 每次重置到不同状态

m =

     4


n =

     5


x =

    0.1673    0.5880    0.8256    0.1117    0.4950
    0.8620    0.1548    0.7900    0.1363    0.1476
    0.9899    0.1999    0.3185    0.6787    0.0550
    0.5144    0.4070    0.5341    0.4952    0.8507
    0.8843    0.7487    0.0900    0.1897    0.5606


x =

    0.9296    0.8790    0.6126    0.8013    0.5747
    0.6967    0.9889    0.9900    0.2278    0.8452
    0.5828    0.0005    0.5277    0.4981    0.7386
    0.8154    0.8654    0.4795    0.9009    0.5860


x =

    0.2467    0.6609    0.7690    0.0170    0.8449
    0.6664    0.7298    0.5814    0.1209    0.2094
    0.0835    0.8908    0.9283    0.8627    0.5523
    0.6260    0.9823    0.5801    0.4843    0.6299


x =

    0.0320    0.0495    0.1231
    0.6147    0.4896    0.2055
    0.3624    0.1925    0.1465


resultofRAND1 =

    0.1891


resultofRAND2 =

    0.8301
    0.6705
    0.0845
    0.0686
    0.0371
    0.3854
    0.1653
    0.3752
    0.7297
    0.4534
    0.8596
    0.5685
    0.9848
    0.3742
    0.3715
    0.9499
    0.9774
    0.7428
    0.4958
    0.4157
    0.0777
    0.3299
    0.9429
    0.0906
    0.3091
    0.5518
    0.0350
    0.0018
    0.9854
    0.8229
    0.4586
    0.9710
         0
         0
    0.0000

2)标准正态分布随机函数

m = 4
n = 5
x = randn(n) %生成n*n随机矩阵,其元素在0-1内.
x = randn(m,n) %生成m*n随机矩阵.
x = randn([m,n]) %生成m*n随机矩阵
% x = randn(m,n,p...) %生成m*n*p*...随机矩阵
% x = randn([m n p ... ]) %生成m*n*p*...随机矩阵
x = randn(size(A)) %生成与矩阵A相同大小的随机矩阵.
resultofRAND1 = randn  %无变量输入时只产生一个随机数
resultofRAND2 = randn('state')  %产生包括均衡发生器当前状态的35个随机数
% resultofRAND3 = randn('state',s) %状态重置为s
% resultofRAND4 = randn('state',0) %重置发生器到初始状态
% resultofRAND5 = randn('state',j) %对整数j重置发生器到第j个状态
% resultofRAND6 = randn('state',sum(100*clock)) % 每次重置到不同状态

m =

     4


n =

     5


x =

   -1.7254   -0.0022    0.9608    0.3763   -0.5100
    0.2882    0.0931    1.7382   -0.2270   -1.3216
   -1.5942   -0.3782   -0.4302   -1.1489   -0.6361
    0.1102   -1.4827   -1.6273    2.0243    0.3179
    0.7871   -0.0438    0.1663   -2.3595    0.1380


x =

   -0.7107   -0.4256    1.0635   -0.3712   -0.5568
    0.7770    1.0486    1.1569   -0.7578   -0.8951
    0.6224    0.6607    0.0530   -0.5640   -0.4093
    0.6474    2.5088   -1.2884    0.5551   -0.1609


x =

    0.4093    1.3244   -1.3853   -0.8927   -0.2269
   -0.9526   -0.2132    0.3105    1.9085   -0.1625
    0.3173   -0.1345   -0.2495    0.1222    0.6901
    0.0780   -1.1714    0.5037    1.0470    0.5558


x =

   -1.1203   -1.4158   -0.3680
   -1.5327    0.0596   -1.3610
   -1.0979   -0.4113    0.7796


resultofRAND1 =

    0.4394


resultofRAND2 =

   362436069
   521288629

3)正态分布随机函数

% x = normrnd (MU,SIGMA) 返回均值为MU,标准差为SIGMA的正态分布随机数据,x可以是矩阵或者向量.
% x = normrnd (MU,SIGMA,m) m指定随机数的个数.
% x = normrnd (MU,SIGMA,m,n) m,n表示行数和列数.

% 产生10个均值为2,方差5的随机数.
x = normrnd (2,sqrt(5),1,10)
x = 2 + sqrt(5)*randn(1,10)

x =

    1.7996    4.2834    0.0457    2.9273    2.7791    2.7810    0.3694    2.7308    0.8487   -0.0045


x =

   -0.6906    4.3206    0.1084    1.6134   -0.7026    1.3356   -5.2271   -0.4305   -1.1896   -0.2684

六,符号运算

1.符号表达式的生成

1)用单引号生成符号表达式 (表达式,方程,微分方程)

f = 'exp(x)'
f = 'a*x^2+bx+c=0'
f = 'D2y - 2Dy - 3y =0'

f =

exp(x)


f =

a*x^2+bx+c=0


f =

D2y - 2Dy - 3y =0

2)使用函数sym来生成符号表达式 ,使用符号数组

A = sym('[a b c ; e f g ]')
f = sym('ax +b =0')

 
A =
 
[ a, b, c]
[ e, f, g]
 
 
f =
 
ax + b = 0
 

3)使用符号表达式来生成, 但是不能生成符号方程.

syms  y u;
p = exp(-y/u)

 
p =
 
1/exp(y/u)
 

2.符号表达式的计算

1)提取分子,分母[n,d]=numden(a) n = numden(a)

f = sym ('a*x^2/(b-x)')
[n,d] = numden(f)

 
f =
 
(a*x^2)/(b - x)
 
 
n =
 
a*x^2
 
 
d =
 
b - x
 

2)符号表达式的基本运算

B = sym('x+1')
C = sym('x^2-1')
D = B + C

 
B =
 
x + 1
 
 
C =
 
x^2 - 1
 
 
D =
 
x^2 + x
 

3.符号表达式的高级运算

% compose(f,g)
% compose(f,g,z)
% compose(f,g,x,z)
% compose(f,g,x,y,z)

syms x,y;
f = 1/x^3
g = tan(y)
compose(f,g)

% 任意建立矩阵A,然后找出在[10 ,20 ]区间的元素的位置.
a=[10 11 8 9 20 44 40]
ElementBetween10and20 = a(a>=10 & a<=20)

 
f =
 
1/x^3
 
 
g =
 
tan(y)
 
 
ans =
 
1/tan(y)^3
 

a =

    10    11     8     9    20    44    40


ElementBetween10and20 =

    10    11    20


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