Lab 2.2

Block Matrices and Graphs

Lab goal

Explain how np.block joins compatible rectangles and why one entry of a squared adjacency matrix counts length-two walks.

Lecture-note concepts

Code extensionThe full notebook continues to Schur complements, block inverses, longer walks, and bipartite reordering.

Key ideas

Compatible blocksX = [[A, B], [C, D]]: Blocks in one block row need the same height; blocks in one block column need the same width.
Block multiplicationtop-left = A E + B G: The usual row-by-column rule still applies when each entry is itself a compatible matrix.
Adjacency power(adj²)ᵢⱼ = # two-step walks: Entry (i,j) of G² counts the length-two walks from vertex i to vertex j.

Lab 2.2 · Guided example

This trace keeps two ideas from the notebook—block arithmetic and adjacency powers—and removes the larger inverse and 8-vertex examples.

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

Simplified fromCh2-2 Block Matrices & Graphs.ipynb ↗The values and graph are intentionally smaller than the full notebook examples.

L01 / 20Join four compatible blocks

line action

import numpy as np

All traced lines

Predict

The four definitions become one 3×3 matrix.

Before

Matrix multiplication combines compatible rows and columns, whether we view them as blocks or graph connections.

row × column → one output entry

After

Reveal to compare.

X.shapenot stored ?A complete square matrix.

Continue in Python

Run the original code and change the inputs.