Visualizing Linear Algebra Decompositions

In this notebook we just demonstrate the utility function xyzpy.visualize_matrix on various linear algebra decompositions taken from scipy. This function plots matrices with the values of numbers directly mapped to color. By default, complex phase gives the hue, with

• real positive = blue

• real negative = orange

• imaginary positive = purple

• imaginary negative = green

whereas the magnitude gives the saturation, such that \(|z| \sim 0\) gives white.

%config InlineBackend.figure_formats = ['svg']
import xyzpy as xyz
import numpy as np
import scipy.linalg as sla

rng = np.random.default_rng(42)

First we’ll start with a non-symmetric random matrix with some small complex parts:

X = (
    0.1 * rng.normal(size=(20, 20)) +
    0.01j * rng.normal(size=(20, 20))
)
xyz.visualize_matrix(X, figsize=(2, 2));

Singular Value Decomposition

xyz.visualize_matrix(sla.svd(X), figsize=(8, 6));

The 1D array of real singular values in decreasing magnitude is shown as a diagonal.

Eigen-decomposition

xyz.visualize_matrix(sla.eig(X), figsize=(6, 4));

Here we see the introduction of many complex numbers far from the real axis.

Schur decomposition

xyz.visualize_matrix(sla.schur(X), figsize=(6, 4));

If you look closely here at the color sequence of the left diagonal it follows the eigen decomposition.

xyz.visualize_matrix(sla.schur(X.real), figsize=(6, 4));

QR Decomposition

xyz.visualize_matrix(sla.qr(X), figsize=(6, 4));

Polar Decomposition

xyz.visualize_matrix(sla.polar(X), figsize=(6, 4));

LU Decomposition

xyz.visualize_matrix(sla.lu(X), figsize=(8, 6));

Multiplying the left matrix in reorders the rows of the \(L\) factor:

xyz.visualize_matrix(sla.lu(X, permute_l=True), figsize=(6, 4));