Lernen Matrix Operations in Python

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1. Addition and Subtraction

Two matrices $A$ and $B$ of the same shape can be added:


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import numpy as np

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

C = A + B  
print(f'C:\n{C}')    # C = [[4, 6], [12, 14]]

2. Multiplication Rules

Matrix multiplication is not element-wise.

Rule: if $A$ has shape $(n, m)$ and $B$ has shape $(m, l)$ , then the result has shape $(n, l)$ .


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import numpy as np

# Example random matrix 3x2
A = np.array([[1, 2],
              [3, 4],
              [5, 6]])
print(f'A:\n{A}')

# Example random matrix 2x4 
B = np.array([[11, 12, 13, 14],
              [15, 16, 17, 18]])
print(f'B:\n{B}')

# product shape (3, 4)
product = np.dot(A, B)     
print(f'np.dot(A, B):\n{product}')

# or equivalently
product = A @ B
print(f'A @ B:\n{product}')

3. Transpose

Transpose flips rows and columns.

General rule: if $A$ is $(n \times m)$ , then $A^T$ is $(m \times n)$ .


              1234567
            
import numpy as np

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

A_T = A.T  # Transpose of A
print(f'A_T:\n{A_T}')

4. Inverse of a Matrix

A matrix $A$ has an inverse $A^{-1}$ if:

A \cdot A^{-1} = I

Where $I$ is the identity matrix.

Not all matrices have inverses. A matrix must be square and full-rank.


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import numpy as np

A = np.array([[1, 2],
              [3, 4]])

A_inv = np.linalg.inv(A)   # Inverse of A
print(f'A_inv:\n{A_inv}')

I = np.eye(2)              # Identity matrix 2x2
print(f'A x A_inv = I:\n{np.allclose(A @ A_inv, I)}')  # Check if product equals identity

What is the output of this Python code?

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[[5 12]
 [21 32]]

[[19 22]
 [43 50]]

[[17 20]
 [39 46]]

[[6 8]
 [10 12]]

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Abschnitt 1. Kapitel 31

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Abschnitt 1. Kapitel 31