>> m_full=magic(1100);%Create a 1100*1100 Matrix
>> m_full(m_full>50)=0;%Replace elements greater than 50 with zero
>> m_sparse=sparse(m_full);%Convert this general matrix into a sparse matrix form for storage
>> whos
  Name             Size                Bytes  Class     Attributes

  m_full        1100x1100            9680000  double              
  m_sparse      1100x1100               9608  double    sparse    

m_full and m_sparse are actually matrices, but the size of bytes they occupy depends on the two storage methods.

• Creating Sparse Matrix

Sparse (A) function implements sparse matrix creation

>> A(1,4)=1;A(2,2)=1;A(3,1)=1;A(3,2)=2;A(4,3)=3
A =
     0     0     0     1
     0     1     0     0
     1     2     0     0
     1     0     3     0
>> s=sparse(A)%Will matrix A Converting to Sparse Matrix Storage
s =
   (3,1)        1
   (4,1)        1
   (2,2)        1
   (3,2)        2
   (4,3)        3
   (1,4)        1
>> B=full(s)%Converting Sparse Matrix to General Matrix
B =
     0     0     0     1
     0     1     0     0
     1     2     0     0
     1     0     3     0

1. Creating Matrix Directly

sparse(i，j,s,m,n)

I, j are row number and column number, s i s a vector of all non-zero elements, m i s row number, n i s column number

>> S=sparse([1,3,2,1,4],[3,1,4,1,4],[1,2,3,4,5],4,4)
S =
   (1,1)        4
   (3,1)        2
   (1,3)        1
   (2,4)        3
   (4,4)        5
>> full(S)
ans =
     4     0     1     0
     0     0     0     3
     2     0     0     0
     0     0     0     5

2. Create sparse matrices from diagonal elements

Use spdiags (B, d, m, n)

This function is used to create a sparse matrix with m rows and n columns. Its non-zero elements come from the elements in matrix B and are arranged diagonally. The parameter d specifies the diagonal position of sparse matrix B in matrix B. It can be considered that the principal diagonal of matrix B is the 0 diagonal line, which adds one bit to the upper right and subtracts one to the lower left.

>> B=reshape(1:12,4,3);
>> d=[-3,0,2];
>> A=spdiags(B,d,7,4)

A =

   (1,1)        5
   (4,1)        1
   (2,2)        6
   (5,2)        2
   (1,3)       11
   (3,3)        7
   (6,3)        3
   (2,4)       12
   (4,4)        8
   (7,4)        4

>> full(A)

ans =

     5     0    11     0
     0     6     0    12
     0     0     7     0
     1     0     0     8
     0     2     0     0
     0     0     3     0
     0     0     0     4

3. Operation of Sparse Matrix

Combination of Sparse Matrix

A =

     1     0     0
     0     0     1
     1     2     0

>> B=sparse(A)

B =

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

>> C=[A(:,1),B(:,1)]

C =

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

So C is a sparse matrix.

Assignment of Sparse Matrix Submatrices

>> A=[1 0 0;0 0 1;1 2 0];
>> B=sparse(A);
>> C=sparse(cat(1,full(B),A))

C =

   (1,1)        1
   (3,1)        1
   (4,1)        1
   (6,1)        1
   (3,2)        2
   (6,2)        2
   (2,3)        1
   (5,3)        1

>> i=[1 2 3]

i =

     1     2     3

>> j=[1 2 3]

j =

     1     2     3

>> T=C(i,j)

T =

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

>> C(j,i)=T

C =

   (1,1)        1
   (3,1)        1
   (4,1)        1
   (6,1)        1
   (3,2)        2
   (6,2)        2
   (2,3)        1
   (5,3)        1

Assigning sparse matrices to general matrices still returns sparse matrices

 

4. Exchange and reordering of sparse matrices

For commutative matrix p, row commutation for sparse matrix S can be expressed as p*S and column commutation as S*p.

For a vector p, P is a general vector, row exchange for sparse matrix is S(p,:), column exchange is S (:, p), or row exchange for a column, such as S(p, n).

>> A
A =
     1     0     0
     0     0     1
     1     2     0
>> p=[1 3 2]; 
>> A(p,:)
ans =
     1     0     0
     1     2     0
     0     0     1

Actually, it's equivalent to index.

>> s_1=speye(3)
s_1 =
   (1,1)        1
   (2,2)        1
   (3,3)        1
>> p_1=s_1(p,:)%Row-to-row swapping of matrices
p_1 =
   (1,1)        1
   (3,2)        1
   (2,3)        1

Return still sparse matrix

>> A=[0  1 2 3;3 2 1 0;0 0 2 0;1 0 0 1]
A =

     0     1     2     3
     3     2     1     0
     0     0     2     0
     1     0     0     1

>> P=colperm(A)

P =

     1     2     4     3

>> B=A(:,P)

B =

     0     1     3     2
     3     2     0     1
     0     0     0     2
     1     0     1     0

The colperm() function is used to return the ranking of the two numbers of lungs in column vectors in A, and the sorted matrix is obtained by exchange.