Lab 2.3

Testing LU Decomposition

Lab goal

See what SciPy returns, why $Q = P$ .T gives $Q \mathbin{\mathtt{@}} A = L \mathbin{\mathtt{@}} U$ , and where rank deficiency appears.

Lecture-note concepts

Code extensionThe full notebook generates random 10×11 matrices, repeats the test ten times, and plots the collected results.

Key ideas

Rank bound $A = X Y \Rightarrow \operatorname{rank}(A) \le 2$: A product with a two-dimensional middle space cannot have rank larger than two.
Pivoted LU $A = P L U \Longleftrightarrow Q A = L U$: SciPy returns $A = P L U$ . Setting $Q = P$ .T gives the course form $Q A = L U$ .
Residual $\Vert Q A - L U\Vert$: The residual compresses the reconstruction difference into one nonnegative number.

Lab 2.3 · Guided example

The notebook’s repeated random experiment is reduced to one deterministic 3×4 matrix so the rank and reconstruction are visible without a histogram.

Select a line to see its action, array shape, and concrete operation.

Simplified fromCh2-3 Test LU-Decomposition.ipynb ↗The full source repeats this test with larger unseeded random matrices.

L01 / 12Create two narrow factors

line action

import numpy as np

All traced lines

Predict

X has two columns and Y has two rows.

Before

We will multiply a 3×2 matrix by a 2×4 matrix, forcing the result through a two-dimensional middle space.

(3×2)(2×4) → 3×4 with rank at most 2

After

Reveal to compare.

Continue in Python

Run the original code and change the inputs.