Matrices for Networks - StudyPulse
StudyPulse Logo
Sign Up
Boost Your VCE Scores Today with StudyPulse
8000+ Questions AI Tutor Help
Create Free Account Learn More
Home Subjects General Mathematics Matrices in networks

Matrices for Networks

General Mathematics

About these notes

Area Of Study

Matrices

Unit

Unit 3: General Mathematics Unit 3

Join StudyPulse

Sign up for a free account to access more study resources and track your progress.

Create Free Account Learn More
Get started in 60 seconds
StudyPulse

Matrices for Networks

General Mathematics
01 May 2026

Matrices for Networks and Transitions

Adjacency Matrices

A network (graph) can be represented as an adjacency matrix. For \(n\) vertices, the matrix is \(n \times n\) where entry \(a_{ij}\) = number of edges from vertex \(i\) to vertex \(j\).

Example Network

Consider 3 towns: A, B, C with roads: A–B, A–C, B–C, B–B (loop).

\[\text{Adjacency matrix} = \begin{array}{c|ccc} & A & B & C \\ \hline A & 0 & 1 & 1 \\ B & 1 & 1 & 1 \\ C & 1 & 1 & 0 \end{array}\]
  • Diagonal entry \(a_{BB} = 1\) because there is a loop at B.
  • The matrix is symmetric for undirected graphs.

Counting Walks

The \((i,j)\) entry of \(A^n\) gives the number of walks of length \(n\) from vertex \(i\) to vertex \(j\).

\[A^2 = A \times A \quad \Rightarrow \quad (A^2)_{ij} = \text{number of 2-step walks from } i \text{ to } j\]

Transition Matrices

A transition matrix \(T\) models movement between states over time. Each column sums to 1 (column-stochastic).

\[T = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}\]

Column 1: From State 1, probability 0.7 of staying, 0.4 of moving to State 2 (columns sum to 1.1 — this example is for illustration; valid matrices have columns summing to 1).

State Vector

The state vector \(S_n\) gives the distribution across states at step \(n\):

\[S_n = T \times S_{n-1} = T^n \times S_0\]

Worked Example

At week 0: 80% of customers use Brand X, 20% use Brand Y.

\[S_0 = \begin{pmatrix} 0.8 \\ 0.2 \end{pmatrix}, \quad T = \begin{pmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{pmatrix}\]

Week 1:

\[S_1 = TS_0 = \begin{pmatrix} 0.9(0.8)+0.2(0.2) \\ 0.1(0.8)+0.8(0.2) \end{pmatrix} = \begin{pmatrix} 0.76 \\ 0.24 \end{pmatrix}\]

76% Brand X, 24% Brand Y after one week.

KEY TAKEAWAY: Adjacency matrices encode graph structure; transition matrices encode probabilistic movement. Both rely on matrix multiplication to compute multi-step results.

VCAA FOCUS: Be able to set up a transition matrix from a verbal description, identify the state vector, and compute \(S_1\) or \(S_2\) by hand.

Table of Contents