The simplest way to construct a small vector or matrix is to use brackets:
>> [1 2 3 5] % a row vector (actually, a 1x4 matrix)
ans =
1 2 3 5
>> [1, 2, 3, 5] % a row vector using commas
ans =
1 2 3 5
>> [1; 2; 3; 5] % a column vector (actually, a 4x1 matrix)
ans =
1
2
3
5
>> [1 2 3 4; 2 4 6 8; 3 6 9 12] % a 3x4 matrix using semicolons
ans =
1 2 3 4
2 4 6 8
3 6 9 12
>> [1 2 3 4 % a 3x4 matrix using Enter
2 4 6 8
3 6 9 12]
ans =
1 2 3 4
2 4 6 8
3 6 9 12
>> 3 % a scalar (actually, a 1x1 matrix)
ans =
3
A space or comma separates entries within a column, and a semicolon or Enter separates rows.
A few things to note:
All the examples above have scalar entries. It is also possible to have matrix entries:
A = [1 2; 3 4];
[A [5; 6]]
ans =
1 2 5
3 4 6
In this case, all matrices in the same row must have the same column dimension, and all matrices in the same column must have the same row dimension. Unfortunately, this is one place where scalars are not expanded. For example, to append a column of ones, you must do the following:
>> [A 1]
??? Error using ==> horzcat
All matrices on a row in the bracketed expression must have the
same number of rows.
>> [A ones(size(A,1), 1)] % [A ones(2,1)] works as well
ans =
1 2 1
3 4 1
Notes: ones(2, 1) creates a 2×1 matrix of 1s. size( A, 1 ) returns the number of rows of A and size( A, 2 ) returns the number of columns of A.
The colon operator x:y generates a row vector (1xn matrix) of the form [x, x+1, x+2, ..., x+n] where x+n <= y and x+n+1 > y.
>> 3:7
ans =
3 4 5 6 7
>> 1.2:7.3
ans =
1.2000 2.2000 3.2000 4.2000 5.2000 6.2000 7.2000
The three variable form x:dx:z generates row vector [x, x+dx, x+2*dx, ..., x+n*dx] where x+n*dx <= y and x+(n+1)*dx > y.
>> 3:2:11
ans =
3 5 7 9 11
>> 1.2:1.25:7.3
ans =
1.2000 2.4500 3.7000 4.9500 6.2000
The colon operator can be used in conjunction with brackets:
>> [1:3 5 7:10]
ans =
1 2 3 5 7 8 9 10
General matrix and vector constructors are:
The diag(v) function returns a matrix with specified diagonal entries.
>> diag([1 2 3 4])
ans =
1 0 0 0
0 2 0 0
0 0 3 0
0 0 0 4
The diag(v, n) shifts the diagonal either to the right (positive n) or down (negative n.) diag(v, 0) is equivalent to diag(v).
.
>> diag([1 2 3], 2)
ans =
0 0 1 0 0
0 0 0 2 0
0 0 0 0 3
0 0 0 0 0
0 0 0 0 0
>> diag([1 2 3], -2)
ans =
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 2 0 0 0
0 0 3 0 0
An alternate use of the diag function is to return the diagonal of a matrix.
Short for identity, eye returns an identity matrix.
>> eye(3)
ans =
1 0 0
0 1 0
0 0 1
>> eye(3,4)
ans =
1 0 0 0
0 1 0 0
0 0 1 0
The linspace(x, y, N) function generates a row vector with N linearly spaced points between x and y.
>> linspace(3, 7, 10) % 10 linearly spaced points between 3 and 7
ans =
3.0000 3.4444 3.8889 4.3333 4.7778 5.2222 5.6667 6.1111 6.5556 7.0000
The function linspace(x, y, N) is equivalent to x:(y-x)/(N-1):y.
The logspace(x, y, N) function generates a row vector with N logarithmically spaced points between 10x and 10y.
>> logspace(.2, 1.2, 10) % 10 logarithmically spaced points between 100.2 and 101.2
ans =
1.5849 2.0470 2.6438 3.4145 4.4101 5.6958 7.3564 9.5012 12.2713 15.8489
The ones(n) function generates an nxn matrix of ones.
>> ones(3)
ans =
1 1 1
1 1 1
1 1 1
>> ones(3, 4)
ans =
1 1 1 1
1 1 1 1
1 1 1 1
The rand(n) function generates an nxn matrix of random values uniformly selected from the open interval (0, 1).
>> rand(3)
ans =
0.0362 0.4919 0.2617
0.2929 0.6533 0.5844
0.8158 0.0433 0.1370
>> rand(3, 4)
ans =
0.4541 0.1952 0.0997 0.6332
0.4029 0.7694 0.0950 0.3427
0.2528 0.9234 0.9740 0.9991
The zeros(n) function generates an nxn matrix of zeros.
>> zeros(3)
ans =
0 0 0
0 0 0
0 0 0
>> zeros(3, 4)
ans =
0 0 0 0
0 0 0 0
0 0 0 0
| Function | Comments |
|---|---|
| compan | compan(A) return the companion matrix of A |
| hadamard | hadamard(n) return the Hadamard matrix associated with the integer n |
| hankel | hankel(v) return the Hankel matrix with v as the first column u as the last row |
| hilb | hilb(n) return an nxn Hilbert matrix A where A(i,j) = 1/(i+j-1) |
| invhilb | invhilb(n) returns the inverse of hilb(n) |
| magic | magic(n) return an n × n magic square |
| pascal | pascal(n) return an n × n matrix with entries from Pascal's triangle |
| rosser | rosser return the 8 × 8 rosser matrix |
| toeplitz | toeplitz(c, r) return a Toeplitz matrix with first column equal to c and first row equal to r. |
| vander | vander(v) for a vector of size n return an n × n matrix whose columns are powers (.^)of the vector v. That is, A(i,j) = v(i)^(n-j). |
| wilkinson | wilkinson(n) return an n × n tridiagonal Wilkinson matrix. |