Matrix Operations with Matlab

In this lesson, I will explain how to perform matrix operations using Matlab.

First, create a square matrix M1 with two rows and two columns.

>> M1=[1 4;2 3]
M 1 =
1 4
2 3

Next, create another square matrix M2 with two rows and two columns.

>> M2=[3 1;7 5]
M2 =
3 1
7 5

Now, with these two matrices M1 and M2, let's go through some practical examples of matrix calculations.

Matrix Addition

To perform matrix addition, use the plus operator (+).

Type M1+M2

>> M1+M2
ans =
4 5
9 8

$$ M1 + M2 = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix} = \begin{pmatrix} 1+3 & 4+1 \\ 2+7 & 3+5 \end{pmatrix} = \begin{pmatrix} 4 & 5 \\ 9 & 8 \end{pmatrix} $$

Matrix Subtraction

To perform matrix subtraction, use the minus operator (-).

Type M1-M2

>> M1-M2
ans =
-2 3
-5 -2

$$ M1 - M2 = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} - \begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix} = \begin{pmatrix} 1-3 & 4-1 \\ 2-7 & 3-5 \end{pmatrix} = \begin{pmatrix} -2 & 3 \\ -5 & -2 \end{pmatrix} $$

Matrix Multiplication

To perform matrix multiplication, use the multiplication operator (*).

Type M1*M2

>> M1*M2
ans =
31 21
27 17

$$ M1 \cdot M2 = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix} = \begin{pmatrix} 1 \cdot 3 + 4 \cdot 7 & 1 \cdot 1 + 4 \cdot 5 \\ 2 \cdot 3 + 3 \cdot 7 & 2 \cdot 1 + 3 \cdot 5 \end{pmatrix} = \begin{pmatrix} 31 & 21 \\ 27 & 17 \end{pmatrix} $$

Note that matrix multiplication is called row-by-column multiplication.

You can only perform matrix multiplication between two matrices if the number of columns in the first matrix (M1) is equal to the number of rows in the second matrix (M2).

Element-wise Matrix Multiplication

Element-wise matrix multiplication calculates the product of elements that are in the same position.

It is a different type of matrix multiplication compared to row-by-column multiplication.

To perform element-wise multiplication, use the dot multiplication operator (.*).

>> M1 .* M2
ans =
3 4
14 15

In element-wise multiplication, both matrices must have the same number of rows and columns.

$$ M1 \ .* \ M2 = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \ .* \ \begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix} = \begin{pmatrix} 1 \cdot 3 & 4 \cdot 1 \\ 2 \cdot 7 & 3 \cdot 5 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 14 & 15 \end{pmatrix} $$

Multiplying a matrix by a scalar

To calculate the product of a matrix and a scalar, use the multiplication operator (*).

For example, enter 2*M1

>> 2*M1
ans =
2 8
4 6

The elements of the matrix are multiplied by the scalar number 2.

$$ 2 \cdot M1 = 2 \cdot \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 2 \cdot 1 & 2 \cdot 4 \\2 \cdot 2 & 2 \cdot 3 \end{pmatrix} = \begin{pmatrix} 2 & 8 \\4 & 6 \end{pmatrix} $$

Matrix division

Matrix division can be achieved by multiplying the first matrix by the inverse matrix of the second, M1·M2^-1.

To calculate the division of two matrices in Matlab, enter M1*inv(M2)

>> M1*inv(M2)
ans =
-2.87500 1.37500
-1.37500 0.87500

Alternatively, you can also enter M1/M2

In this case, Matlab automatically performs the inversion of the second matrix.

>> M1/M2
ans =
-2.87500 1.37500
-1.37500 0.87500

The final result is always the same.

Element-wise matrix division

Element-wise division calculates the quotient between elements that are in the same position.

It is another type of matrix division.

To perform element-wise division, use the operator ./

>> M1 ./ M2
ans =
0.33333 4.0000
0.28571 0.6000

In the case of element-wise division, the two matrices must have the same number of rows and columns.

$$ M1 \ ./ \ M2 = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \ ./ \ \begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix} = \begin{pmatrix} \frac{1}{3} & \frac{4}{1} \\ \frac{2}{7} & \frac{3}{5} \end{pmatrix} = \begin{pmatrix} 0.33333 & 0.28571 \\ 4 & 0.6 \end{pmatrix} $$

Dividing a matrix by a scalar

Dividing a matrix by a scalar is done using the division operator (/).

For example, to divide matrix M1 by two, enter M1/2

>> M1/2
ans =
0.50000 2.00000
1.00000 1.50000

All elements of matrix M1 are divided by the scalar number 2.

$$ \frac{M1}{2} = \frac{1}{2} \cdot \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \cdot 1 & \frac{1}{2} \cdot 4 \\ \frac{1}{2} \cdot 2 & \frac{1}{2} \cdot 3 \end{pmatrix} = \begin{pmatrix} 0.5 & 2 \\ 1 & 1.5 \end{pmatrix} $$

Element-wise matrix exponentiation

To raise every element of a matrix to the same

For example, to raise the elements of matrix M1 to the power of 2, type M1.^2

>> M1.^2
ans =
1 16
4 9

$$ M1 \ \text{.^} \ 2 = \begin{pmatrix} 1^2 & 4^2 \\ 2^2 & 3^2 \end{pmatrix} = \begin{pmatrix} 1 & 16 \\ 4 & 9 \end{pmatrix} $$

Matrix determinant

Matlab has a specific function to calculate the determinant of a square matrix. It is the det() function.

For example, to calculate the determinant of matrix M1, type det(M1)

>> det(M1)
ans = -5

$$ \text{det} (M1) = det \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} = 1 \cdot 3 - 4 \cdot 2 = -5 $$

Matrix rank

To find the rank of a matrix, use the rank() function.

For example, to calculate the rank of matrix M1, type rank(M1)

>> rank(M1)
ans = 2

The rank is equal to 2 because the determinant of the 2x2 matrix is not zero. $$ \text{det} (M1) = det \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} = 1 \cdot 3 - 4 \cdot 2 = -5 $$

Matrix trace

To calculate the trace of a matrix, use the trace() function.

For example, to calculate the trace of matrix M1, type trace(M1)

>> trace(M1)
ans = 4

The trace of a matrix is equal to the sum of the elements on the main diagonal. $$ \text{trace} (M1) = \text{trace} \begin{pmatrix} \color{red}1 & 4 \\ 2 & \color{red}3 \end{pmatrix} = 1 + 3 = 4 $$

Transpose of a Matrix

To transpose the rows and columns of a matrix, use the transpose() function.

For example, to transpose matrix M1, type transpose(M1)

>> transpose(M1)
ans =
1 2
4 3

Alternatively, you can use the matrix transpose operator by adding an apostrophe after the name of the matrix.

>> M1'
ans =
1 2
4 3

In a transpose of matrix, the rows of the matrix become columns and vice versa. $$ \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix}^T = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix} $$

Matrix inverse

To calculate the inverse of a matrix, use the inv() function.

For example, to calculate the inverse of matrix M1, type inv(M1)

>> inv(M1)
ans =
-0.60000 0.80000
0.40000 -0.20000

The inverse of matrix M1 is a matrix that, when multiplied by M1, results in an identity matrix. An identity matrix is a matrix with elements equal to 1 on the main diagonal and 0 elsewhere. $$ M1 \cdot \text{inv} (M1) = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} -0.6 & 0.8 \\ 0.4 & -0.2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$

The Characteristic Polynomial

To calculate the characteristic polynomial of a square matrix, you can use the poly() function.

>> poly(M1)
ans =
1 -4 -5