with python

import numpy as np

create matrix

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

multiply @ (__matmul__)
transpose .T

result1 = A.T@B
result2 = A@B.transpose()

det det(A)

det_A = np.linalg.det(A) # -2.0

• for
  • cramer’s rule (solving equation)
  • det(A) = 0; singular
    inverse

A_inv = np.linalg.inv(A)

rules
identity matrix (for i,j => aij = 1)

identity = A@A_inv # I 
# or 
identity = np.identity(2)

matrix transformation

• transformation matrix (T)
• TA = B
• Geometric transformation

eigenvector

• (v)
• non zero vector,
• applied linear transformation (matrix mul) => “direction” unchanged (but length can be changed)
  eigenvalue
• lambda (λ)
• the amount it it is stretched, squished
  eigendecompose
  jacobian

resources

• numpy for matrices and vectors
  • https://www2.lawrence.edu/fast/GREGGJ/Python/numpy/numpyLA.html
• Machine Learning & Linear Algebra — Eigenvalue and eigenvector
  • https://jonathan-hui.medium.com/machine-learning-linear-algebra-eigenvalue-and-eigenvector-f8d0493564c9

created on: Wed Jul 08 2026