Matrices are fundamental to MATLAB. Therefore, we need to become familiar with matrix generation and manipulation. Matrices can be generated in several ways.

A vector is a special case of a matrix. The purpose of this section is to show how to create vectors and matrices in MATLAB. As discussed earlier, an array of dimension 1 × n is called a row vector, whereas an array of dimension m × 1 is called a column vector. The elements of vectors in MATLAB are enclosed by square brackets and are separated by spaces or by commas. For example, to enter a row vector, v, type

v = [1 4 7 10 13]
v =
1 4 7 10 13
Column vectors are created in a similar way, however, semicolon (;) must separate the components of a column vector,
w = [1;4;7;10;13]
w =
1
4
7
10
13
On the other hand, a row vector is converted to a column vector using the transpose operator. The transpose operation is denoted by an apostrophe or a single quote (').
w = v'
w =
1
4
7
10
13
Thus, v(1) is the first element of vector v, v(2) its second element, and so forth.

Furthermore, to access blocks of elements, we use MATLAB’s colon notation (:). For example, to access the first three elements of v, we write,

v(1:3)
ans =
1 4 7
Or, all elements from the third through the last elements,
v(3,end)
ans =
7 10 13
where end signifies the last element in the vector. If v is a vector, writing
v(:)
produces a column vector, whereas writing
v(1:end)
produces a row vector.

A matrix is an array of numbers. To type a matrix into MATLAB you must begin with a square bracket, [
• separate elements in a row with spaces or commas (,)
• use a semicolon (;) to separate rows
• end the matrix with another square bracket, ].
Here is a typical example. To enter a matrix A, such as,
1 2 3
A = 4 5 6
7 8 9
type,

A = [1 2 3; 4 5 6; 7 8 9]
MATLAB then displays the 3 × 3 matrix as follows,
A =
1 2 3
4 5 6
7 8 9

We select elements in a matrix just as we did for vectors, but now we need two indices. The element of row i and column j of the matrix A is denoted by A(i,j). Thus, A(i,j) in MATLAB refers to the element A of matrix A. The first index is the row number and the second index is the column number. For example, A(1,3) is an element of first row and third column. Here, A(1,3)=3.

The colon operator will prove very useful and understanding how it works is the key to efficient and convenient usage of MATLAB. It occurs in several different forms. Often we must deal with matrices or vectors that are too large to enter one element at a time. For example, suppose we want to enter a vector x consisting of points (0,0.1,0.2,0.3,...,5). We can use the command

x = 0:0.1:5;
The row vector has 51 elements.

On the other hand, there is a command to generate linearly spaced vectors: linspace. It is similar to the colon operator (:), but gives direct control over the number of points. For example,
y = linspace(a,b)
generates a row vector y of 100 points linearly spaced between and including a and b.
y = linspace(a,b,n)
generates a row vector y of n points linearly spaced between and including a and b. This is useful when we want to divide an interval into a number of subintervals of the same length.

The colon operator can also be used to pick out a certain row or column. For example, the statement A(m:n,k:l) specifies rows m to n and column k to l. Subscript expressions refer to portions of a matrix. For example,

A(2,:)
ans =
4 5 6
is the second row elements of A.
The colon operator can also be used to extract a sub-matrix from a matrix A.
A(:,2:3)
ans =
2 3
5 6
8 0
A(:,2:3) is a sub-matrix with the last two columns of A.

To extract a submatrix B consisting of rows 2 and 3 and columns 1 and 2 of the matrix A, do the following

B = A([2 3],[1 2])
B =
4 5
7 8
To interchange rows 1 and 2 of A, use the vector of row indices together with the colon operator.
C = A([2 1 3],:)
C =
4 5 6
1 2 3
7 8 0
It is important to note that the colon operator (:) stands for all columns or all rows.

To delete a row or column of a matrix, use the empty vector operator, [ ].

A(3,:) = []
A =
1 2 3
4 5 6
Third row of matrix A is now deleted. To restore the third row, we use a technique for creating a matrix
A = [A(1,:);A(2,:);[7 8 0]]
A =
1 2 3
4 5 6
7 8 0
Matrix A is now restored to its original form.

To determine the dimensions of a matrix or vector, use the command size. For example,

size(A)
ans =
3 3
means 3 rows and 3 columns.
Or more explicitly with,
[m,n]=size(A)

The transpose operation is denoted by an apostrophe or a single quote ('). It flips a matrix about its main diagonal and it turns a row vector into a column vector. Thus,

A'
ans =
1 4 7
2 5 8
3 6 0
By using linear algebra notation, the transpose of m × n real matrix A is the n × m matrix that results from interchanging the rows and columns of A. The transpose matrix is denoted AT.

MATLAB provides functions that generates elementary matrices. The matrix of zeros, the matrix of ones, and the identity matrix are returned by the functions zeros, ones, and eye, respectively.
Table 2.4: Elementary matrices
eye(m,n) Returns an m-by-n matrix with 1 on the main diagonal
eye(n) Returns an n-by-n square identity matrix
zeros(m,n) Returns an m-by-n matrix of zeros
ones(m,n) Returns an m-by-n matrix of ones
diag(A) Extracts the diagonal of matrix A
rand(m,n) Returns an m-by-n matrix of random numbers

MATLAB provides a number of special matrices (see Table 2.5). These matrices have interesting properties that make them useful for constructing examples and for testing algorithms.