Algebraic Operations on a Matrix in R

In this article, we will discuss how we perform Algebraic Operations on a Matrix in R Programming Language. so we will start with the matrix.

What is Matrix?

A Matrix is a rectangular arrangement of numbers in rows and columns. In a matrix, as we know rows are the ones that run horizontally and columns are the ones that run vertically. In R Programming Language matrices are two-dimensional, homogeneous data structures. These are some examples of matrices.

What are the Algebraic Operations?

Basic algebraic operations are any one of the traditional operations of arithmetic, which are addition, subtraction, multiplication, division, raising to an integer power, and taking roots. These operations may be performed on numbers, in which case they are often called arithmetic operations. We can perform many more algebraic operations on a matrix in R. Algebraic operations that can be performed on a matrix in R:

• Operations on a single matrix
• Unary operations
• Binary operations
• Linear algebraic operations
  • Rank, determinant, transpose, inverse, trace of a matrix
  • Nullity of a matrix
  • Eigenvalues and eigenvectors of matrices
  • Solve a linear matrix equation

Operations on a single matrix

We can use overloaded arithmetic operators to do element-wise operation on a matrix to create a new matrix. In case of +=, -=, *= operators, the existing matrix is modified. 

# R program to demonstrate
# basic operations on a single matrix

# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(a)

# add 1 to every element
cat("Adding 1 to every element:\n")
print(a + 1)

# subtract 3 from each element
cat("Subtracting 3 from each element:\n")
print(a-3)

# multiply each element by 10
cat("Multiplying each element by 10:\n")
print(a * 10)

# square each element
cat("Squaring each element:\n")
print(a ^ 2)

# modify existing matrix
cat("Doubled each element of original matrix:\n")
print(a * 2)

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    4    5    6
[3, ]    7    8    9
Adding 1 to every element:
     [, 1] [, 2] [, 3]
[1, ]    2    3    4
[2, ]    5    6    7
[3, ]    8    9   10
Subtracting 3 from each element:
     [, 1] [, 2] [, 3]
[1, ]   -2   -1    0
[2, ]    1    2    3
[3, ]    4    5    6
Multiplying each element by 10:
     [, 1] [, 2] [, 3]
[1, ]   10   20   30
[2, ]   40   50   60
[3, ]   70   80   90
Squaring each element:
     [, 1] [, 2] [, 3]
[1, ]    1    4    9
[2, ]   16   25   36
[3, ]   49   64   81
Doubled each element of original matrix:
     [, 1] [, 2] [, 3]
[1, ]    2    4    6
[2, ]    8   10   12
[3, ]   14   16   18

Unary operations

Many unary operations can be performed on a matrix in R. This includes sum, min, max, etc. 

# R program to demonstrate
# unary operations on a matrix

# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(a)

# maximum element in the matrix
cat("Largest element is:\n")
print(max(a))

# minimum element in the matrix
cat("Smallest element is:\n")
print(min(a))

# sum of element in the matrix
cat("Sum of elements is:\n")
print(sum(a))

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    4    5    6
[3, ]    7    8    9
Largest element is:
[1] 9
Smallest element is:
[1] 1
Sum of elements is:
[1] 45

Binary operations

These operations apply on a matrix elementwise and a new matrix is created. You can use all basic arithmetic operators like +, -, *, /, etc. In case of +=, -=, = operators, the existing matrix is modified. 

# R program to demonstrate
# binary operations on a matrix

# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(a)

# Create another 3x3 matrix
b = matrix(
c(1, 2, 5, 4, 6, 2, 9, 4, 3), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The another 3x3 matrix:\n")
print(b)

cat("Matrix addition:\n")
print(a + b)

cat("Matrix subtraction:\n")
print(a-b)

cat("Matrix element wise multiplication:\n")
print(a * b)

cat("Regular Matrix multiplication:\n")
print(a %*% b)

cat("Matrix elementwise division:\n")
print(a / b)

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    4    5    6
[3, ]    7    8    9
The another 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    5
[2, ]    4    6    2
[3, ]    9    4    3
Matrix addition:
     [, 1] [, 2] [, 3]
[1, ]    2    4    8
[2, ]    8   11    8
[3, ]   16   12   12
Matrix subtraction:
     [, 1] [, 2] [, 3]
[1, ]    0    0   -2
[2, ]    0   -1    4
[3, ]   -2    4    6
Matrix element wise multiplication:
     [, 1] [, 2] [, 3]
[1, ]    1    4   15
[2, ]   16   30   12
[3, ]   63   32   27
Regular Matrix multiplication:
     [, 1] [, 2] [, 3]
[1, ]   36   26   18
[2, ]   78   62   48
[3, ]  120   98   78
Matrix elementwise division:
          [, 1]      [, 2] [, 3]
[1, ] 1.0000000 1.0000000  0.6
[2, ] 1.0000000 0.8333333  3.0
[3, ] 0.7777778 2.0000000  3.0

Linear algebraic operations

One can perform many linear algebraic operations on a given matrix In R. Some of them are as follows:

Rank, determinant, transpose, inverse, trace of a matrix

• Rank: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it is the dimension of the column space (or equivalently, the row space) of the matrix.
• Determinant: The determinant has various geometric interpretations, such as measuring the volume scaling factor of a linear transformation represented by the matrix.
• Transpose: The transpose of a matrix is obtained by swapping its rows and columns.
• Inverse: The inverse of a square matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix.
• Trace: The trace of a square matrix is the sum of its diagonal elements.

# R program to demonstrate
# Linear algebraic operations on a matrix

# Importing required library
 # For rank of matrix
library(pracma)
# For trace of matrix
library(psych) 

# Create a 3x3 matrix
A = matrix(
c(6, 1, 1, 4, -2, 5, 2, 8, 7), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(A)

# Rank of a matrix
cat("Rank of A:\n")
print(Rank(A))

# Trace of matrix A
cat("Trace of A:\n")
print(tr(A))

# Determinant of a matrix
cat("Determinant of A:\n")
print(det(A))

# Transpose of a matrix
cat("Transpose of A:\n")
print(t(A))

# Inverse of matrix A
cat("Inverse of A:\n")
print(inv(A))

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    6    1    1
[2, ]    4   -2    5
[3, ]    2    8    7
Rank of A:
[1] 3
Trace of A:
[1] 11
Determinant of A:
[1] -306
Transpose of A:
     [, 1] [, 2] [, 3]
[1, ]    6    4    2
[2, ]    1   -2    8
[3, ]    1    5    7
Inverse of A:
            [, 1]         [, 2]        [, 3]
[1, ]  0.17647059 -0.003267974 -0.02287582
[2, ]  0.05882353 -0.130718954  0.08496732
[3, ] -0.11764706  0.150326797  0.05228758

Nullity of a matrix

The nullity of a matrix is the dimension of the null space, also known as the kernel, of the matrix. The null space of a matrix.

# R program to demonstrate
# nullity of a matrix

# Importing required library
library(pracma)

# Create a 3x3 matrix
a = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(a)

# No of column
col = ncol(a)

# Rank of matrix
rank = Rank(a)

# Calculating nullity
nullity = col - rank

cat("Nullity of matrix is:\n")
print(nullity)

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    4    5    6
[3, ]    7    8    9
Nullity of matrix is:
[1] 1

Eigenvalues and eigenvectors of matrices

# R program to illustrate
# Eigenvalues and eigenvectors of metrics

# Create a 3x3 matrix
A = matrix(
c(1, 2, 3, 4, 5, 6, 7, 8, 9), 
nrow = 3,             
ncol = 3,             
byrow = TRUE         
)
cat("The 3x3 matrix:\n")
print(A)

# Calculating Eigenvalues and eigenvectors
print(eigen(A))

Output:

The 3x3 matrix:
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    4    5    6
[3, ]    7    8    9
eigen() decomposition
$values
[1]  1.611684e+01 -1.116844e+00 -1.303678e-15

$vectors
           [, 1]        [, 2]       [, 3]
[1, ] -0.2319707 -0.78583024  0.4082483
[2, ] -0.5253221 -0.08675134 -0.8164966
[3, ] -0.8186735  0.61232756  0.4082483

Solve a linear matrix equation

# R program to illustrate
# Solve a linear matrix equation of metrics

# Importing library for applying pseudoinverse
library(MASS)

# Create a 2x2 matrix
A = matrix(
c(1, 2, 3, 4), 
nrow = 2,             
ncol = 2,             
)
cat("A = :\n")
print(A)

# Create another 2x1 matrix
b = matrix(
c(7, 10), 
nrow = 2,             
ncol = 1,             
)
cat("b = :\n")
print(b)

cat("Solution of linear equations:\n")
print(solve(A)%*% b)

cat("Solution of linear equations using pseudoinverse:\n")
print(ginv(A)%*% b)

Output:

A = :
     [, 1] [, 2]
[1, ]    1    3
[2, ]    2    4
b = :
     [, 1]
[1, ]    7
[2, ]   10
Solution of linear equations:
     [, 1]
[1, ]    1
[2, ]    2
Solution of linear equations using pseudoinverse:
     [, 1]
[1, ]    1
[2, ]    2