clear all;
Basic Matrix manipulation
In this tutorial we will explain some of the basics of working with matrices in matlab.
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Create a Matrix
- Create a matrix structure by using square brackets [ ]
- Separate columns by a space or comma ,
- Separate rows by semicolon ;
M =
syms m11 m12 m13 m21 m22 m23 m31 m32 m33 real M = [m11, m12, m13 ; %row separated by semicolon m21, m22, m23 ; %column separated by comma m31 m32 m33] %column separated by space
$$ \displaystyle \left(\begin{array}{ccc} m_{11} & m_{12} & m_{13} \newline m_{21} & m_{22} & m_{23} \newline m_{31} & m_{32} & m_{33} \end{array}\right) $$
The functions eye() lets you create an identity matrix:
IdentityMatrix = eye(3)
IdentityMatrix = 3x3
1 0 0
0 1 0
0 0 1
To create a matrix of zeros or ones you can use the functions below. You can define the dimensions as (row,column):
OneMatrix = ones(2,4)
OneMatrix = 2x4
1 1 1 1
1 1 1 1
ZeroMatrix = zeros(2,2)
ZeroMatrix = 2x2
0 0
0 0
Transpose a Matrix
Transpose a matrix by extending it with an apostrophe '
M_transposed = M'
$$ \displaystyle \left(\begin{array}{ccc} m_{11} & m_{21} & m_{31} \newline m_{12} & m_{22} & m_{32} \newline m_{13} & m_{23} & m_{33} \end{array}\right) $$
This can also be used to transform a row vector in a column vector. This can be done with a variable or with the array itself.
syms v1 v2 v3 real
V_column = [v1, v2, v3]
$$ \displaystyle \left(\begin{array}{ccc} v_1 & v_2 & v_3 \end{array}\right) $$
V_row = [v1, v2, v3]'
$$ \displaystyle \left(\begin{array}{c} v_1 \newline v_2 \newline v_3 \end{array}\right) $$
V_row_2 = V_column'
$$ \displaystyle \left(\begin{array}{c} v_1 \newline v_2 \newline v_3 \end{array}\right) $$
Invert a Matrix
To invert a matrix we have a few options such as:
Sample_Matrix = [1, 2, 3;
4, 5, 6;
6, 2, 10]
Sample_Matrix = 3x3
1 2 3
4 5 6
6 2 10
Sample_Matrix_inv_1 = inv(Sample_Matrix)
Sample_Matrix_inv_1 = 3x3
-1.0556 0.3889 0.0833
0.1111 0.2222 -0.1667
0.6111 -0.2778 0.0833
Sample_Matrix_inv_2 = Sample_Matrix^-1
Sample_Matrix_inv_2 = 3x3
-1.0556 0.3889 0.0833
0.1111 0.2222 -0.1667
0.6111 -0.2778 0.0833
Sample_Matrix_inv_3 = eye(3)/Sample_Matrix
Sample_Matrix_inv_3 = 3x3
-1.0556 0.3889 0.0833
0.1111 0.2222 -0.1667
0.6111 -0.2778 0.0833
Invert a non square Matrix
To invert a non square Matrix you can use the pseudo inverse:
$$ A^{\dagger} ={\left(A^T \cdot A\right)}^{-1} \cdot A^T $$
Sample_Matrix_non_square = Sample_Matrix(1:3,1:2)
Sample_Matrix_non_square = 3x2
1 2
4 5
6 2
Sample_Matrix_non_square_inverse_1 = inv(Sample_Matrix_non_square' * Sample_Matrix_non_square) * Sample_Matrix_non_square'
Sample_Matrix_non_square_inverse_1 = 2x3
-0.0590 -0.0641 0.2192
0.1214 0.2175 -0.1653
Sample_Matrix_non_square_inverse_2 = pinv(Sample_Matrix_non_square)
Sample_Matrix_non_square_inverse_2 = 2x3
-0.0590 -0.0641 0.2192
0.1214 0.2175 -0.1653
Access Matrix Elements
To access a specific elements of a matrix you can include (row,column) as an extension of the variable:
M(1,1)
You can also access sections of your matrix if you define a region as (start_row : end_row, start_column : end_column)
M(1:2,1:2)
$$ \displaystyle \left(\begin{array}{cc} m_{11} & m_{12} \newline m_{21} & m_{22} \end{array}\right) $$
With this Notation we can also write into specific elements of the matrix and override them:
M(1:3, 3)=V_row
$$ \displaystyle \left(\begin{array}{ccc} m_{11} & m_{12} & v_1 \newline m_{21} & m_{22} & v_2 \newline m_{31} & m_{32} & v_3 \end{array}\right) $$
M(1:2,1:2)=eye(2)
$$ \displaystyle \left(\begin{array}{ccc} 1 & 0 & v_1 \newline 0 & 1 & v_2 \newline m_{31} & m_{32} & v_3 \end{array}\right) $$
Combining Matrices
To combine two 2D matrices into one Matrix we can use the cat(Dimension, Matrix_1, ..., Matrix_n) function:
Matrix_1 = ones(3)
Matrix_1 = 3x3
1 1 1
1 1 1
1 1 1
Matrix_2 = ones(3) * 2
Matrix_2 = 3x3
2 2 2
2 2 2
2 2 2
Matrix_combined_1 = cat(1, Matrix_1, Matrix_2)
Matrix_combined_1 = 6x3
1 1 1
1 1 1
1 1 1
2 2 2
2 2 2
2 2 2
Matrix_combined_2 = cat(2, Matrix_1, Matrix_2)
Matrix_combined_2 = 3x6
1 1 1 2 2 2
1 1 1 2 2 2
1 1 1 2 2 2
Matrix_combined_3 = cat(3, Matrix_1, Matrix_2)
Matrix_combined_3 =
Matrix_combined_3(:,:,1) =
1 1 1
1 1 1
1 1 1
Matrix_combined_3(:,:,2) =
2 2 2
2 2 2
2 2 2
Matrix_3 = ones(3) * 3;
Matrix_combined_4 = cat(3, Matrix_1, Matrix_2, Matrix_3)
Matrix_combined_4 =
Matrix_combined_4(:,:,1) =
1 1 1
1 1 1
1 1 1
Matrix_combined_4(:,:,2) =
2 2 2
2 2 2
2 2 2
Matrix_combined_4(:,:,3) =
3 3 3
3 3 3
3 3 3
Mathematical Operations with Matrices
To add two matrices of the same size:
Matrix_4 = [1, 2, 3;
1, 2, 3;
1, 2, 3]
Matrix_4 = 3x3
1 2 3
1 2 3
1 2 3
Matrix_added = Matrix_1 + Matrix_4
Matrix_added = 3x3
2 3 4
2 3 4
2 3 4
To subtract matrices of the same size:
Matrix_subtracted = Matrix_4 - Matrix_1
Matrix_subtracted = 3x3
0 1 2
0 1 2
0 1 2
To multiply two matrices:
Matrix_multiply = Matrix_2 * Matrix_4
Matrix_multiply = 3x3
6 12 18
6 12 18
6 12 18
To have element-wise operations like
$$ \left\lbrack \begin{array}{cc} \textrm{m11} & \textrm{m12}\newline \textrm{m21} & \textrm{m22} \end{array}\right\rbrack =\left\lbrack \begin{array}{cc} \textrm{a11}\cdot \;\textrm{b11} & \textrm{a12}\cdot \textrm{b12}\newline \textrm{a21}\cdot \textrm{b21} & \textrm{a22}\cdot \textrm{b22} \end{array}\right\rbrack =\left\lbrack \begin{array}{cc} \textrm{a11} & \textrm{a12}\newline \textrm{a21} & \textrm{a22} \end{array}\right\rbrack \ldotp \times \left\lbrack \begin{array}{cc} \textrm{b11} & \textrm{b12}\newline \textrm{b21} & \textrm{b22} \end{array}\right\rbrack $$
we can add a . in front of the operator:
Matrix_element_multiply = Matrix_2 .* Matrix_4
Matrix_element_multiply = 3x3
2 4 6
2 4 6
2 4 6
Matrix_element_devision = Matrix_4 ./ Matrix_2
Matrix_element_devision = 3x3
0.5000 1.0000 1.5000
0.5000 1.0000 1.5000
0.5000 1.0000 1.5000
Matrix_element_squared = Matrix_4.^2
Matrix_element_squared = 3x3
1 4 9
1 4 9
1 4 9
Other useful Functions
Dimensions
To get the dimensions of a Matrix you can use the size() function:
[rows, columns, dimensions] = size(Matrix_combined_4)
rows = 3
columns = 3
dimensions = 3
if you are only interested in some of the dimensions:
[rows, ~, ~] = size(Matrix_4)
rows = 3
Summing of Matrix elements
You can sum matrix column elements by using the sum() function:
Matrix_column_sum = sum(Matrix_4)
Matrix_column_sum = 1x3
3 6 9
you can add all the elements of a Matrix to each other by giving the option input "all":
Matrix_total_sum = sum(Matrix_4,"all")
Matrix_total_sum = 18
Maximum/Minimum
You can extract the maximum or minimum value of each column by using the max() or min() function
Max_column = max(Matrix_4)
Max_column = 1x3
1 2 3
Min_column = min(Matrix_4);
To get the maximum or minimum of the entire matrix you can give the optional input [] and "all"
MaxAll = max(Matrix_4, [], "all")
MaxAll = 3
MinAll = min(Matrix_4, [], "all")
MinAll = 1
Rank
Using the rank() function you can extract the rank of you matrix, which is defined as the number of linearly independent columns (or rows) in that matrix. The rank provides insight into the dimensionality of the column space of the matrix.
For an identity matrix the rank is the same as its rows/columns. The Identity matrix is "full rank".
IdentityRank = rank(eye(4))
IdentityRank = 4
However for the Matrix 4, which contains the same row multiple times the rank is only 1.
MatrixRank = rank(Matrix_4)
MatrixRank = 1