MATLABit
MATLAB stands for MATrix LABoratory. It’s a powerful programming language and software tool created by MathWorks. Its extensive application across engineering, scientific research, academic instruction, and algorithmic design stems from its strengths in numerical computation, data analysis, graphical visualization, and simulation. With a foundation in matrix algebra, MATLAB efficiently manages large datasets and complex mathematical models. So, let's commence to know multiply arrays in MATLAB.
Table of Contents
Introduction
Multiplication of arrays in MATLAB is one of the most important operations for scientific and engineering applications. Since MATLAB was originally designed as a Matrix Laboratory in the late 1970s by Cleve Moler, special attention has been given to efficient and intuitive handling of matrix and array computations. Unlike some other programming languages where multiplication is limited to scalars, MATLAB provides two distinct but related forms of multiplication: matrix multiplication and element-wise multiplication.
Matrix Multiplication (*)
The operator * in MATLAB follows the rules of linear algebra.
This means that if A is an m × n matrix and B
is an n × r matrix, then their product A * B results
in an m × r matrix. Each element of the result is obtained by
computing the dot product of a row of A with a column of B.
For example:
A = [2 4;
1 3;
0 5];
B = [3 1;
2 6];
C = A * B
% The result C is a 3 × 2 matrix:
C = [ (2*3 + 4*2) (2*1 + 4*6);
(1*3 + 3*2) (1*1 + 3*6);
(0*3 + 5*2) (0*1 + 5*6) ]
C = [ 14 26;
9 19;
10 30 ]
Element-Wise Multiplication (.*)
When two arrays of the same size are multiplied using the operator
.*, MATLAB performs element-wise multiplication.
In this operation, each component of one array is multiplied by the
equivalent component of the other array. This is extremely useful in
numerical computing, data analysis, and image processing.
Example:
X = [4 7 2];
Y = [3 5 2];
Z = X .* Y
Z = [ (4*3) (7*5) (2*2) ]
Z = [ 12 35 4 ]
Scalar Multiplication
MATLAB also allows direct multiplication of an array by a scalar. In this case, each element of the array is scaled by the scalar value.
M = [1 -2 4;
0 5 3];
2 * M
ans = [ 2 -4 8;
0 10 6 ]
Historical Note
Multiplication of matrices is at the heart of linear algebra, which itself
serves as the foundation for numerical computing. MATLAB’s design philosophy ensures
that both classical matrix multiplication and element-wise operations are
easy to perform, without requiring loops. This clear distinction between
* and .* reflects MATLAB’s emphasis on combining
mathematical precision with programming convenience.
In summary, multiplication of arrays in MATLAB can be carried out in three
ways: matrix multiplication (*), element-wise multiplication
(.*), and scalar multiplication. Each serves a different
purpose, but together they make MATLAB a powerful environment for
computational mathematics.
Multiplication of Arrays in MATLAB
In MATLAB, the multiplication operator * is executed according
to the norms of linear algebra. This means that if
P and B are two matrices, the operation
P * B can only be performed when the number of columns in
P are same as the number of rows in B.
The result will then be a new matrix with the equivalent rows as
P and the equivalent columns as B.
Matrix Multiplication Dimensions
For example, if P is a 4 × 3 matrix and B is a
3 × 2 matrix, then the product P * B will be a 4 × 2 matrix.
Each element of the result is obtained as the dot product of a row of
P with a column of B:
(P11*B11 + P12*B21 + P13*B31) (P11*B12 + P12*B22 + P13*B32)
(P21*B11 + P22*B21 + P23*B31) (P21*B12 + P22*B22 + P23*B32)
(P31*B11 + P32*B21 + P33*B31) (P31*B12 + P32*B22 + P33*B32)
(P41*B11 + P42*B21 + P43*B31) (P41*B12 + P42*B22 + P43*B32)
Numerical Example
P = [ -1 0 2;
7 4 3;
6 0 8;
9 2 5 ]; % Define a 4×3 matrix P
B = [ 3 6;
2 0;
4 7 ]; % Define a 3×2 matrix B
C = P * B
The result is:
C =
(-1*3 + 0*2 + 2*4) (-1*6 + 0*0 + 2*7)
(7*3 + 4*2 + 3*4) (7*6 + 4*0 + 3*7)
(6*3 + 0*2 + 8*4) (6*6 + 0*0 + 8*7)
(9*3 + 2*2 + 5*4) (9*6 + 2*0 + 5*7)
C =
5 8
41 63
50 92
49 89
Non-Commutativity
It is essential to keep that in mind that matrix multiplication is
not commutative. In other words, A * B
does not necessarily equal B * A. In fact, in the example above,
trying B * A produces an error since the dimensions are
not compatible (B has 2 columns, while A has 4 rows).
Multiplying Square Matrices
F = [ 2 4;
1 3 ];
G = [ 5 7;
0 6 ];
F * G
The result is:
ans =
(2*5 + 4*0) (2*7 + 4*6)
(1*5 + 3*0) (1*7 + 3*6)
ans =
10 38
5 25
G * F
The result is different:
ans =
(5*2 + 7*1) (5*4 + 7*3)
(0*2 + 6*1) (0*4 + 6*3)
ans =
17 41
6 18
This confirms that F * G ≠ G * F.
Vector Multiplication
Two vectors can be multiplied if they have the same number of elements, however, one must be expressed as a horizontal vector, while the other should be represented as a vertical vector:
AV = [ 4 2 5 ]; % Horizontal vector
BV = [ -2;
-3;
0]; % Vertical vector
AV * BV % Row × Column → Scalar (dot product)
ans = -14
BV * AV % Column × Row → 3×3 matrix
ans =
-8 -4 -10
-12 -6 -15
0 0 0
Scalar Multiplication
When an array is multiplied by a scalar (a single number, treated as a 1 × 1 array), every element in the array is multiplied by that scalar:
A = [ 3 0 2 -1;
1 4 8 6;
9 0 3 2 ]; % Define a 3×4 matrix
b = 4;
b * A
The result is:
ans =
12 0 8 -4
4 16 32 24
36 0 12 8
Connection to Systems of Linear Equations
Linear algebra rules of array multiplication provide a convenient way of expressing systems of equations. For instance, the system:
2x1 + 3x2 + 4x3 = 15
1x1 + 5x2 + 2x3 = 20
3x1 + 0x2 + 6x3 = 25
can be written as:
[ 2 3 4 ] [ x1 ] [ 15 ]
[ 1 5 2 ] * [ x2 ] = [ 20 ]
[ 3 0 6 ] [ x3 ] [ 25 ]
or more compactly in matrix notation as: A * X = B.
Applications
Array multiplication is one of the most powerful tools in MATLAB, especially because it follows the rules of linear algebra. It is widely applied in scientific computing, engineering, data analysis, and machine learning. Since MATLAB is designed for matrix-based computations, multiplication is at the heart of most real-world applications.
1. Solving Systems of Linear Equations
Many real-world problems can be expressed as a system of linear equations. Using matrix multiplication, these systems can be written compactly as A × X = B, where A is a coefficient matrix, X is the vector of unknowns, and B is the constants vector. MATLAB efficiently solves such systems using matrix operations instead of handling each equation individually.
A = [3 2 1; 4 5 6; 7 8 9]; % Coefficient matrix
X = [x1; x2; x3]; % Unknowns
B = [12; 30; 45]; % Constants
% System representation: A * X = B
2. Computer Graphics and Transformations
In graphics and visualization, transformations such as rotation, scaling, and translation are performed using matrix multiplication. For example, a 2D point or an image can be rotated around the origin using a rotation matrix multiplied by the vector of coordinates.
theta = pi/4; % Rotation angle (45 degrees)
R = [cos(theta) -sin(theta);
sin(theta) cos(theta)];
P = [5; 2]; % A point in 2D space
NewP = R * P; % Rotated coordinates
3. Signal Processing
In digital signal processing (DSP), array multiplication is used for filtering, convolution, and Fourier transforms. By multiplying signals with transformation matrices, MATLAB helps in analyzing signals in time and frequency domains.
4. Machine Learning and Artificial Intelligence
Neural networks, regression models, and optimization algorithms rely heavily on array multiplication. Weight matrices in machine learning models are multiplied with input data arrays to generate predictions. MATLAB's matrix operations make training and testing computational models efficient.
5. Engineering Applications
In electrical, mechanical, and civil engineering, MATLAB uses array multiplication for structural analysis, circuit design, and control systems. For instance, state-space representations in control systems are solved directly using matrix multiplication.
6. Economics and Data Analysis
In finance and economics, matrix multiplication is used for portfolio optimization, input-output models, and economic forecasting. Large datasets can be processed quickly through vectorized operations, making MATLAB ideal for quantitative research.
Summary
From solving equations and designing engineering systems to training AI models and simulating graphics, array multiplication in MATLAB provides a universal and efficient framework for computation. It not only simplifies mathematical operations but also ensures high performance when working with large-scale problems.
Conclusion
Array multiplication in MATLAB is not just a mathematical operation, but a foundation for solving complex computational problems. Unlike element-wise operations, matrix multiplication strictly follows the norms of linear algebra and ensuring mathematical consistency.
Through examples, we have seen that multiplication can be performed between matrices, vectors, and scalars, each following specific dimension rules. While matrix-by-matrix multiplication allows us to handle systems of equations and transformations, scalar multiplication scales every element of an array, and vector multiplication provides dot and cross products that are widely used in geometry and physics.
One of the most important takeaways is that matrix multiplication is not commutative, meaning A × B ≠ B × A in most cases. This property highlights the need for careful attention to dimensions and order of operations when performing calculations.
Overall, array multiplication provides a powerful framework for applications in engineering, computer science, data analysis, economics, graphics, and machine learning. Mastering this concept in MATLAB allows users to efficiently model, analyze, and solve real-world problems with accuracy and speed.
In short: Understanding and applying matrix multiplication in MATLAB equips learners and professionals with one of the most essential tools in numerical computing.
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