Diagonal Matrices¶

The np.diag function provides bidirectional conversion between diagonal components and diagonal matrices.

Core Concept¶

np.diag works in two directions depending on input dimensionality.

\[ \text{diagonal components} \quad\stackrel{\text{np.diag}}{\longleftrightarrow}\quad \text{diagonal matrix} \]

Components to Matrix¶

Convert a 1D array of diagonal values into a 2D diagonal matrix.

1. Basic Conversion¶

import numpy as np

def main():
    a = np.diag([1, 2, 3])
    print("np.diag([1, 2, 3])")
    print(a)

if __name__ == "__main__":
    main()

Output:

np.diag([1, 2, 3])
[[1 0 0]
 [0 2 0]
 [0 0 3]]

2. Input Requirements¶

The input must be a 1D array or list of diagonal values.

Diagonal Offset¶

The k parameter shifts the diagonal position.

1. Main Diagonal¶

import numpy as np

def main():
    a = np.diag([1, 2, 3])
    print("np.diag([1, 2, 3])")
    print(a)

if __name__ == "__main__":
    main()

Default k=0 places values on the main diagonal.

2. Upper Diagonals¶

import numpy as np

def main():
    a = np.diag([1, 2, 3], k=1)
    print("np.diag([1, 2, 3], k=1)")
    print(a)

    a = np.diag([1, 2, 3], k=2)
    print("np.diag([1, 2, 3], k=2)")
    print(a)

if __name__ == "__main__":
    main()

Output:

np.diag([1, 2, 3], k=1)
[[0 1 0 0]
 [0 0 2 0]
 [0 0 0 3]
 [0 0 0 0]]

np.diag([1, 2, 3], k=2)
[[0 0 1 0 0]
 [0 0 0 2 0]
 [0 0 0 0 3]
 [0 0 0 0 0]
 [0 0 0 0 0]]

3. Lower Diagonals¶

import numpy as np

def main():
    a = np.diag([1, 2, 3], k=-1)
    print("np.diag([1, 2, 3], k=-1)")
    print(a)

    a = np.diag([1, 2, 3], k=-2)
    print("np.diag([1, 2, 3], k=-2)")
    print(a)

if __name__ == "__main__":
    main()

Negative k shifts the diagonal below the main diagonal.

Matrix to Components¶

Extract diagonal values from a 2D matrix.

1. Extract Main Diagonal¶

import numpy as np

def main():
    A = np.array([[1, 4, 0],
                  [0, 2, 5],
                  [0, 0, 3]])

    print(f'{np.diag(A) = }')

if __name__ == "__main__":
    main()

Output:

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

2. Extract Off-Diagonals¶

import numpy as np

def main():
    A = np.array([[1, 4, 0],
                  [0, 2, 5],
                  [0, 0, 3]])

    print(f'{np.diag(A, k=1) = }')
    print(f'{np.diag(A, k=-1) = }')

if __name__ == "__main__":
    main()

Output:

np.diag(A, k=1) = array([4, 5])
np.diag(A, k=-1) = array([0, 0])

Visual Geometry¶

Diagonal matrices have a distinctive visual pattern.

1. Visualization Code¶

import numpy as np
import matplotlib.pyplot as plt

def main():
    a = np.diag(range(15))

    fig, ax = plt.subplots(figsize=(4, 4))
    ax.imshow(a, cmap='Blues')
    ax.axis('off')
    plt.show()

if __name__ == "__main__":
    main()

2. Pattern Recognition¶

Non-zero values appear only along a single diagonal line.

Linear Algebra Role¶

Diagonal matrices have special computational properties.

1. Easy Inversion¶

The inverse of a diagonal matrix is simply the reciprocal of each diagonal element.

2. Eigenvalues¶

The eigenvalues of a diagonal matrix are its diagonal elements.

3. Matrix Powers¶

\(D^n\) is computed by raising each diagonal element to the \(n\)-th power.