linear algebra

The linear algebra that you will definitely learn at a science university is summarized in an easy-to-understand and logical manner. By the way, I implemented it in Python. Occasionally, it may be implemented in Julia. .. .. ・ Learn by running with Python! New Mathematics Textbook-Basic Knowledge Required for Machine Learning / Deep Learning- ・ World Standard MIT Textbook Strang Linear Algebra Introduction Understand linear algebra based on and implement it in python.

environment

・ Jupyter Notebook ・ Language: Python3, Julia 1.4.0

Identity matrix

E = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix},\begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}

It is a matrix in which 1s are lined up diagonally from the upper left to the lower right, and the other elements are 0.

Even if the nth-order square matrix is multiplied by the identity matrix with the same number of elements, the output values are the same. Furthermore, the order of products does not matter.

AE = \begin{pmatrix}a & b\\c & d\end{pmatrix}\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}=\begin{pmatrix}a & b\\c & d\end{pmatrix}=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\begin{pmatrix}a & b\\c & d\end{pmatrix}=EA

Python code for identity matrix

import numpy as np

# print("2 × 2 identity matrix")
print(np.eye(2))
# print("The 3x3 identity matrix")
print(np.eye(3))
# print("The 4x4 identity matrix")
print(np.eye(4))

#Implementation
2 × 2 identity matrix
[[1. 0.]
 [0. 1.]]
The 3x3 identity matrix
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
The 4x4 identity matrix
[[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]

Julia code of identity matrix

using LinearAlgebra
Matrix{Int}(I, 2, 2) 
Matrix{Int}(I, 3, 3) 
Matrix{Int}(I, 4, 4)

2×2 Array{Int64,2}:
 1  0
 0  1

3×3 Array{Int64,2}:
 1  0  0
 0  1  0
 0  0  1

4×4 Array{Int64,2}:
 1  0  0  0
 0  1  0  0
 0  0  1  0
 0  0  0  1

Inverse matrix

The inverse matrix is

AA^{-1} = A^{-1}A = E

It has a relationship of. To give an example

A =\begin{pmatrix}1 & 1 \\1 & 2\end{pmatrix},
B =\begin{pmatrix}2 & -1 \\-1 & 1\end{pmatrix}

Find the product of AB when there are two matrices.

AB =\begin{pmatrix}1 & 1 \\1 & 2\end{pmatrix}\begin{pmatrix}2 & -1 \\-1 & 1\end{pmatrix}=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}

Than this

AB = E = BA \Longrightarrow B = A^{-1}

Determinant

A = \begin{pmatrix}a & b\\c & d\end{pmatrix}
\Longleftrightarrow
\begin{vmatrix}A\end{vmatrix}= det A = ad-bc

Furthermore, in the case of a 2x2 matrix,

A^{-1} = \frac{1}{ad-bc}\begin
{pmatrix}d & -b\\-c & a\end{pmatrix}

Can be expressed as, at this time,

ad-bc = 0 \Longleftrightarrow A^{-1}Does not exist
\\
ad-bc \neq 0 \Longleftrightarrow A^{-1}Exists.

Can be written as.

Inverse Matrix Python code including determinant

import numpy as np


a = [input().split() for i in range(2)]
#Enter numbers
#=>1 2
#=>3 4

a = np.array(a)
print(a)
#=>[['1' '2']
#   ['3' '4']]

a = a.astype(np.float64)
print(np.linalg.det(a))
#=>-2.0000000000000004

if np.linalg.det(a) != 0:
    print(np.linalg.inv(a))
else:
    print("No exist")
#=>[[-2.   1. ]
#   [ 1.5 -0.5]]

Actually, even if you calculate as it is with the np.linalg.inv () function without using the if statement, an error will occur if the inverse matrix does not exist. However, I want to use the determinant this time, so I wrote it using the np.linalg.det () function.

Julia code for the inverse matrix including the determinant

using LinearAlgebra
A = [1 2; 3 4]
if det(A) == 0
    print("No exist")
else
    inv(A)
end

#=>2×2 Array{Float64,2}:
# -2.0   1.0
#  1.5  -0.5

The inverse matrix is immediately obtained by the ʻinv () function`.

Finally

Next, I'm going to do linear transformation, so if you want to make the graph stronger. By all means.

Identity matrix and inverse matrix: Linear algebra in Python <4>