Lecture 7: Grids of Numbers

Matrices: organized collections of numbers that encode structure and transformation.

The Hook

Consider this rectangular array:

\begin{pmatrix} 0.96 & -0.24 \\ 0.00 & 0.93 \\ 0.00 & 0.00 \end{pmatrix}

What is it?

Interpretation 1: Two column vectors — the O-H bond vectors in water.
Interpretation 2: Three row vectors — the x, y, z coordinates of the bonds.
Interpretation 3: A transformation that maps something in R² to something in R³.

Same grid of numbers. Three different meanings. This is a matrix.

Recognition

ARRANGEMENT: "Order matters."

A matrix is an organized collection of numbers where:

Position matters (row i, column j)
The whole structure has meaning beyond individual entries
Operations on the structure encode relationships

When to Use Matrices

Situation	Matrix gives you
"I have several vectors to track"	Store as columns (or rows)
"I need to transform vectors"	Linear transformation Av
"I have a system of linear equations"	Ax = b
"I need to rotate/reflect/scale"	Transformation matrix
"I have relational data"	Adjacency matrix
"I'm doing quantum mechanics"	Operators as matrices

Matrix Fundamentals

Definition: An m × n matrix A is a rectangular array of numbers with m rows and n columns.

The entry in row i, column j is denoted $a_{ij}$ or $A_{ij}$.

Special Matrices

Special Matrix Types

Matrix Operations

Matrix Multiplication

Matrix Multiplication Visualizer

Highlight entry:

Entry (i,j) of AB = dot product of row i of A with column j of B

Key requirement: Number of columns of A = number of rows of B.

(m × n) · (n × p) = (m × p)

Matrix Multiplication is NOT Commutative

AB vs BA

Matrix A

Matrix B

AB =

BA =

Matrices as Linear Transformations

Transformation Visualizer

Transform:

Matrix:

Composition of Transformations

Transformation Composer

First:

Then:

Combined matrix = Second × First:

The Inverse Matrix

Definition: For a square matrix A, the inverse A⁻¹ (if it exists) satisfies:

A^{-1}A = AA^{-1} = I

2×2 Inverse Calculator

Matrix A

det(A) = ad - bc = 1

A⁻¹ =

The 2×2 inverse formula:

\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Chemistry Connection: Molecular Coordinates

A molecule with N atoms can be stored as a 3 × N matrix — each column is one atom's position.

Molecular Coordinate Transformer

Molecule:

Rotate z: 0°

Rotation matrix R_z(θ) transforms all atom positions simultaneously.

Chemistry Connection: Hückel Theory

For π-electron systems, the Hamiltonian in the basis of p-orbitals is encoded as a matrix where diagonal = α (orbital energy) and off-diagonal = β (interaction).

Hückel Matrix Builder

Molecule:

Hückel matrix (α = 0, β = -1):

Summary: Matrix Toolkit

Operation	Notation	Result Dimensions
Addition	A + B	Same as A, B
Scalar mult	cA	Same as A
Transpose	A^T	Swap m ↔ n
Multiplication	AB	(m×n)(n×p) → m×p
Inverse	A⁻¹	Same as A (square)

Exercises

Exercise 1: Matrix Operations

Let A = [[1,2],[3,4]] and B = [[5,6],[7,8]].

(a) Compute AB. (b) Compute BA. (c) Verify AB ≠ BA.

(a) AB = [[1·5+2·7, 1·6+2·8], [3·5+4·7, 3·6+4·8]] = [[19, 22], [43, 50]]

(b) BA = [[5·1+6·3, 5·2+6·4], [7·1+8·3, 7·2+8·4]] = [[23, 34], [31, 46]]

Exercise 2: Inverse

Find the inverse of A = [[4,3],[3,2]]. Verify AA⁻¹ = I.

det(A) = 4·2 - 3·3 = 8 - 9 = -1

A⁻¹ = (1/-1)[[2,-3],[-3,4]] = [[-2,3],[3,-4]]

Check: AA⁻¹ = [[4,3],[3,2]][[-2,3],[3,-4]] = [[-8+9, 12-12],[−6+6, 9−8]] = [[1,0],[0,1]] ✓

Exercise 3: Transformation Composition

(a) Write the matrix for reflection across y-axis.

(b) Write the matrix for rotation by 90°.

(a) Reflect y: [[-1,0],[0,1]]

(b) Rotate 90°: [[0,-1],[1,0]]

This is reflection across y = -x.

Exercise 4: Hückel Matrix

For allyl radical (3 carbons in a row), write the Hückel matrix with α=0, β=-1. What is its trace?

H = [[0,-1,0],[-1,0,-1],[0,-1,0]]

Trace = 0 + 0 + 0 = 0 = 3α ✓