Matrices and Matrix Arithmetic

Matrices and Matrix Arithmetic

Matrices are a foundational element of linear algebra. Matrices are used throughout the fi eld of machine learning in the description of algorithms and processes such as the input data variable (X) when training an algorithm.A matrix is a two-dimensional array of scalars with one or more columns and one or more rows.

Defining a Matrix

We can represent a matrix in Python using a two-dimensional NumPy array. A NumPy array can be constructed given a list of lists. For example, below is a 2 row, 3 column matrix.

# create matrix
from numpy import array
A = array([[1, 2, 3], [4, 5, 6]])
print(A)
o/p:
[[1 2 3]
[4 5 6]]
Matrix Addition
# matrix addition
from numpy import array
# define first matrix
A = array([[1, 2, 3],[4, 5, 6]])

print("Matrix A")
print(A)
# define second matrix
B = array([[1, 2, 3],[4, 5, 6]])

print("Matrix B")
print(B)
# add matrices
C = A + B

print("Matrix C")
print(C)
o/p:

Matrix A
[[1 2 3]
[4 5 6]]

Matrix B
[[1 2 3]
[4 5 6]]

Matrix C
[[ 2 4 6]
[ 8 10 12]]
Matrix Subtraction
# matrix subtraction
from numpy import array
# define first matrix
A = array([[1, 2, 3],[4, 5, 6]])

print("Matrix A")
print(A)
# define second matrix
B = array([[0.5, 0.5, 0.5],[0.5, 0.5, 0.5]])

print("Matrix B")
print(B)
# subtract matrices
C = A - B

print("Matrix C")
print(C)
o/p:

Matrix A
[[1 2 3]
[4 5 6]]

Matrix B
[[ 0.5 0.5 0.5]
[ 0.5 0.5 0.5]]

Matrix C
[[ 0.5 1.5 2.5]
[ 3.5 4.5 5.5]]
Matrix Multiplication (Hadamard Product)

Two matrices with the same size can be multiplied together, and this is often called element-wise matrix multiplication or the Hadamard product. It is not the typical operation meant when referring to matrix multiplication.

# matrix Hadamard product
from numpy import array
# define first matrix
A = array([[1, 2, 3],[4, 5, 6]])

print("Matrix A")
print(A)
# define second matrix
B = array([[1, 2, 3],[4, 5, 6]])

print("Matrix B")

print(B)
# multiply matrices
C = A * B

print("Matrix C")
print(C)
o/p:

Matrix A

[[1 2 3]
[4 5 6]]

Matrix B
[[1 2 3]
[4 5 6]]

Matrix C
[[ 1 4 9]
[16 25 36]]
Matrix Division
# matrix division
from numpy import array
# define first matrix
A = array([[1, 2, 3],[4, 5, 6]])

print("Matrix A")
print(A)
# define second matrix
B = array([[1, 2, 3],[4, 5, 6]])

print("Matrix B")
print(B)
# divide matrices
C = A / B

print("Matrix C")
print(C)
o/p:

Matrix A
[[1 2 3]
[4 5 6]]

Matrix B
[[1 2 3]
[4 5 6]]

Matrix C
[[ 1. 1. 1.]
[ 1. 1. 1.]]
Matrix-Matrix Multiplication

Matrix multiplication, also called the matrix dot product is more complicated than the previous

operations and involves a rule as not all matrices can be multiplied together.

C = A.B
or
C = AB
The rule for matrix multiplication is as follows:
The number of columns (n) in the rst matrix (A) must equal the number of rows (m) in
the second matrix (B).
For example, matrix A has the dimensions m rows and n columns and matrix B has the dimensions n and k. The n columns in A and n rows in B are equal. The result is a new matrix with m rows and k columns.
C(m; k) = A(m; n) . B(n; k)
# matrix dot product
import numpy as np
# define first matrix
A =np. array([
[1, 2],
[3, 4],
[5, 6]])

print("Matrix A")
print(A)
# define second matrix
B = np.array([
[1, 2],
[3, 4]])

print("Matrix B")
print(B)
# multiply matrices
C = A.dot(B)

print("Matrix C")
print(C)
# multiply matrices with @ operator
D = A @ B

print("Matrix D")
print(D)

print("Matrix E")

E=np.matmul(A,B)
o/p:

Matrix A
[[1 2]
[3 4]
[5 6]]

Matrix B

[[1 2]
[3 4]]
Matrix C
[[ 7 10]
[15 22]
[23 34]]
Matrix D
[[ 7 10]
[15 22]
[23 34]]

Matrix E

[[ 7 10]
[15 22]
[23 34]]

Matrix-Vector Multiplication

A matrix and a vector can be multiplied together as long as the rule of matrix multiplication is observed. Speci cally, that the number of columns in the matrix must equal the number of items in the vector. As with matrix multiplication, the operation can be written using the dot notation. Because the vector only has one column, the result is always a vector.

c = A . v
# matrix-vector multiplication
from numpy import array
# define matrix
A = array([[1, 2],[3, 4],[5, 6]])

print("Matrix A")
print(A)
# define vector
B = array([0.5, 0.5])

print("Matrix B")
print(B)
# multiply
C = A.dot(B)

print("Matrix C")
print(C)
o/p

Matrix A
[[1 2]
[3 4]
[5 6]]

Matrix B
[ 0.5 0.5]

Matrix C
[ 1.5 3.5 5.5]
Matrix-Scalar Multiplication
A matrix can be multiplied by a scalar. This can be represented using the dot notation between
the matrix and the scalar.
C = A. b
# matrix-scalar multiplication
from numpy import array
# define matrix
A = array([[1, 2], [3, 4], [5, 6]])

print("Matrix A")
print(A)
# define scalar
b = 0.5

print("Scalar b")
print(b)
# multiply
C = A * b

print("Matrix C")
print(C)

o/p:

Matrix A
[[1 2]
[3 4]
[5 6]]

Scalar B
0.5

Matrix C
[[ 0.5 1. ]
[ 1.5 2. ]
[ 2.5 3. ]]