[matlab] matrix

matrix

zeros function

Produce all 0 matrix , Zero matrix .

Invocation format ：

• zeros(m)： produce m×m The zero matrices of .
• zeros(m, n)： produce m×n The zero matrices of .
• zeros(size(A))： Generation and matrix A A zero matrix of the same size .

The following functions are called in the same way .

ones function

Produce all 1 matrix , Unitary matrix .

eye function

The resulting diagonal is 1 Matrix . When the matrix is a square matrix , Get an identity matrix .

rand function

produce (0, 1) Interval uniformly distributed random matrix .

randn function

randn function ： The average production is 0, The variance of 1 Standard normal distribution random matrix .

randi function

randi() Function to generate uniformly distributed pseudo-random integers , The scope is imin–imax, If not specified imin, The default is 1.

r = randi(imax,n)： Generate n×n Matrix ,n If omitted, the 1×1 Matrix

r = randi(imax,m,n)： Generate m×n Matrix

r = randi(imax,[m,n])： ditto , usage ：r = randi(imax,size(A));

r = randi(imax,m,n,p,…)： Generate m×n×p×… Matrix

r = randi(imax,[m,n,p,…])： ditto

r = randi([imin,imax],…)： Specifies the imin, Other parameters are the same as above

application 1

First generation 5 Order two bit random integer matrix A, The resulting mean is 0.6、 The variance of 0.1 Of 5 Order normal distribution random matrix B, The final validation (A+B)I=IA+BI（I Is the unit matrix ）.

• fix(a+(b-a+1)*x)： produce [a, b] Random integers uniformly distributed on an interval .
• $\mu$+$\sigma$x： The mean value is $\mu$、 The variance of ${\sigma }^2$ The random number .

A = fix(10+(99-10+1)*rand(5));
B = 0.6+sqrt(0.1)*randn(5);
C = eye(5);
(A+B)*C == C*A + B*C

give the result as follows ：

ans =

  5×5 logical  Array 

   1   1   1   1   1
   1   1   1   1   1
   1   1   1   1   1
   1   1   1   1   1
   1   1   1   1   1

magic function

n The first order magic cube array consists of 1,2,3,…,n2 common n2 An integer makes up , And every line 、 Each column and main 、 On the sub diagonal n The sum of the two elements is equal .
n The sum of the elements in each row and column of the first-order magic cube array is (1+2+3+…+n2)/n=(n+n3)/2.MATLAB function magic(n) Generate a specific Magic Cube .

M = magic(8);
sum(M(1, :)) % 260
sum(M(:, 1)) % 260

Add up sum Usage of

sum function

Call form ：

• sum(A)： Sum each column , Returns a row vector consisting of the sum of each column .
• sum(A, 1)： And sum(A) Equivalent .
• sum(A, 2)： Sum each line , Returns a column vector consisting of the sum of each row .
• sum(subA)：subA by A The indexer matrix of , Find the sum of all elements in a submatrix .

Usage example ：

A = [1 2 3; 4 5 6;7 8 9];
sum(A) % 12    15    18
sum(A, 1) % 12    15    18
sum(A, 2) 
%     6
%    15
%    24
sum(A(1:2, 2:3)) % 7     9
sum(A(:)) % 45

Adjoint matrix

MATLAB The function that generates the adjoint matrix is compan§, among p Is the coefficient vector of a polynomial , Higher power coefficients rank first , The lower power coefficient comes next .

for example , Generating polynomials $x^3-2x^2-5x+6$ The adjoint matrix of .

p = [1 -2 -5 6];
A = compan(p)

give the result as follows ：

A =

     2     5    -6
     1     0     0
     0     1     0

sparse matrix

• Full storage mode

• Sparse storage

  Sparse storage only stores the values and positions of non-zero elements of the matrix , Line number and column number .

  Be careful , When using sparse storage , The order in which matrix elements are stored does not change , It's also stored in column order .

Conversion between full storage and sparse storage

A=sparse(S)： The matrix S Into a sparse storage matrix A.
S=full(A)： The matrix A A matrix converted to full storage S.

A = sparse(eye(3)) %  Equivalent to  speye(3)
full(A)

give the result as follows ：

A =

   (1,1)        1
   (2,2)        1
   (3,3)        1

ans =

     1     0     0
     0     1     0
     0     0     1

sparse Other call formats for functions

sparse(m, n)： Generate a $m\times n$ A sparse matrix in which all elements of are zero .

A = sparse(4, 5);
A(8) = 3;
A(1, 2) = 4;
full(A(1:3, :))

give the result as follows ：

ans =

     0     4     0     0     0
     0     0     0     0     0
     0     0     0     0     0

First, all zero sparse matrix is generated through the above calling format , Then the assignment and access operations can be carried out through the assignment and access modes of the full storage mode .

---

spase(row, col, val)：row、col and val Is a vector of equal length , Respectively represents the line number , Column number and corresponding value .

A = sparse([1 2 2], [2 1 4], [4 5 -7])
full(A)

give the result as follows ：

A =

   (2,1)        5
   (1,2)        4
   (2,4)       -7

ans =

     0     4     0     0
     5     0     0    -7

spconvert function

The invocation format is ：spconvert(A),A A matrix with three columns for several rows or four columns for several rows , The first column represents the row number of each number , The second column represents the column number of each number , The third column represents the real part of each number , The fourth column represents the imaginary part of each number , If all elements of the matrix are real numbers , The fourth column is not required ; so ,A Each row of the matrix represents a non-zero element , The number of rows is the number of non-zero elements in the sparse matrix .

A = [2 2 1; 2 1 -1; 2 4 3]
spconvert(A)

give the result as follows ：

A =

     2     2     1
     2     1    -1
     2     4     3

ans =

   (2,1)       -1
   (2,2)        1
   (2,4)        3

Banded sparse matrix

[B,d]=spdiags(A)： From banded sparse matrix A Extract all non-zero diagonal elements and assign them to the matrix B And the position vectors of these non-zero diagonals d.
A=spdiags(B,d,m,n)： A sparse storage matrix that generates a banded sparse matrix A, among m、n Is the number of rows and columns of the original banded sparse matrix , matrix B Of the i The column is the... Of the original banded sparse matrix i A non-zero diagonal , vector d Is the position of all non-zero diagonals of the original banded sparse matrix .

A=[11 0 0 12 0 0; 0 21 0 0 22 0; 0 0 31 0 0 32; 41 0 0 42 0 0; 0 51 0 0 52 0]
[B,d]=spdiags(A)

give the result as follows ：

A =

    11     0     0    12     0     0
     0    21     0     0    22     0
     0     0    31     0     0    32
    41     0     0    42     0     0
     0    51     0     0    52     0

B =

     0    11    12
     0    21    22
     0    31    32
    41    42     0
    51    52     0

d =

    -3
     0
     3

spdiags(B, d, 5, 6)

give the result as follows ：

ans =

   (1,1)       11
   (4,1)       41
   (2,2)       21
   (5,2)       51
   (3,3)       31
   (1,4)       12
   (4,4)       42
   (2,5)       22
   (5,5)       52
   (3,6)       32

We can also generate a banded sparse matrix by first generating a fully stored matrix and then converting it to a sparse matrix ：

B = diag([11 21 31 42 52 0]) + diag([12 22 32], 3) + diag([41 51 0], -3);
B = B(1:end-1, :)
sparse(B)

give the result as follows ：

B =

    11     0     0    12     0     0
     0    21     0     0    22     0
     0     0    31     0     0    32
    41     0     0    42     0     0
     0    51     0     0    52     0
     
ans =

   (1,1)       11
   (4,1)       41
   (2,2)       21
   (5,2)       51
   (3,3)       31
   (1,4)       12
   (4,4)       42
   (2,5)       22
   (5,5)       52
   (3,6)       32

First use diag Function to build a square matrix , Add the square matrices to get the completely stored matrix , call sparse Make it sparse .

Be careful ：

spdiags(B,d,m,n) In the format m、n The size relationship of determines how to use the matrix B Fill in ：

Conclusion ：

• When m≤n when , The diagonal below the main diagonal （ That is, the diagonal -1：-n+1） When data filling , from B A column of a matrix Top to bottom DE value ; And the diagonal above the main diagonal （1：n-1） When filling elements , from B A column of From bottom to top DE value .
• When m>n when , When data is filled in the diagonal below the main diagonal , from B A column of a matrix From bottom to top DE value ; When the diagonal above the main diagonal is filled with elements , from B A column of Top to bottom DE value .

See the following example ：

A = reshape(1:15, 5, 3);
B = spdiags(A,[-2 0 2], 5, 5);
C = spdiags(A,[-2 0 2], 5, 6);
D = spdiags(A,[-2 0 2], 5, 4);
A
BB = full(B)
CC = full(C)
DD = full(D)

A =

     1     6    11
     2     7    12
     3     8    13
     4     9    14
     5    10    15


BB =

     6     0    13     0     0
     0     7     0    14     0
     1     0     8     0    15
     0     2     0     9     0
     0     0     3     0    10


CC =

     6     0    11     0     0     0
     0     7     0    12     0     0
     3     0     8     0    13     0
     0     4     0     9     0    14
     0     0     5     0    10     0


DD =

     6     0    13     0
     0     7     0    14
     1     0     8     0
     0     2     0     9
     0     0     3     0

speye function

speye(n)： return n Order unit matrix .

speye(m, n)： return $m\times n$ Unit matrix of .

application

The storage and operation of sparse matrix through sparse storage can improve the efficiency of operation , This is the most important reason why we use sparse matrices .

k1 = [1; 1; 2; 1; 0];
k2 = [2; 4; 6; 6; 1];
k3 = [0; 3; 1; 4; 2];
B = [k1 k2 k3];
d = [-1 0 1];
A = spdiags(B, d, 5, 5);
full(A)
b = [0; 3; 2; 1; 5];
x = A\b

give the result as follows ：

ans =

     2     3     0     0     0
     1     4     1     0     0
     0     1     6     4     0
     0     0     2     6     2
     0     0     0     1     1

x =

   -0.1667
    0.1111
    2.7222
   -3.6111
    8.6111