Module 6 Article: Eigenvalues, Stability, and Diagonalisation

The big idea

An eigenvector of a matrix is a direction that the matrix does not rotate. It only stretches or shrinks that direction. The stretch factor is the eigenvalue.

This matters because eigenvalues tell us how a system behaves:

• whether repeated application of a transformation grows or dies out;
• whether a system is stable;
• how data can be simplified using principal directions;
• how ranking methods such as PageRank work.

Eigenvalues and eigenvectors

For an n x n matrix A, a non-zero vector v is an eigenvector if

Av = lambda v

Here lambda is the eigenvalue.

Rearranging gives

(A - lambda I)v = 0

For a non-trivial solution v != 0, the matrix A - lambda I must be singular. So we solve

det(A - lambda I) = 0

This is the characteristic equation. Its roots are the eigenvalues.

Worked example

Let

A = [[2, 1],
     [1, 2]]

Step 1: Find the eigenvalues

det(A - lambda I) = det([[2-lambda, 1],
                         [1, 2-lambda]])
                  = (2-lambda)^2 - 1
                  = lambda^2 - 4lambda + 3
                  = (lambda - 3)(lambda - 1)

So the eigenvalues are:

• lambda = 3
• lambda = 1

Step 2: Find the eigenvectors

For lambda = 3:

(A - 3I)v = 0
[[ -1, 1],
 [  1,-1]] [x, y]^T = 0

This gives y = x, so one eigenvector is

v1 = [1, 1]^T

For lambda = 1:

(A - I)v = 0
[[1, 1],
 [1, 1]] [x, y]^T = 0

This gives y = -x, so one eigenvector is

v2 = [1, -1]^T

Geometric interpretation

Eigenvectors are the special directions of a transformation.

For the matrix above:

• the direction [1, 1] is stretched by a factor of 3;
• the direction [1, -1] is unchanged because its eigenvalue is 1.

So the matrix acts like a transformation that is simple along those two axes, even if it looks complicated in the standard coordinate system.

Diagonalisation

If a matrix has enough independent eigenvectors, it can be written as

A = P D P^-1

where:

• P contains the eigenvectors as columns;
• D is a diagonal matrix of eigenvalues.

This is useful because powers become easy:

A^k = P D^k P^-1

That is why eigenvalues are important in repeated processes and dynamical systems.

Why eigenvalues matter

• Stability: if the magnitude of an eigenvalue is small, repeated application may shrink toward zero; if it is large, the system may grow.
• PCA: the principal directions of a data set come from the eigenvectors of its covariance matrix.
• PageRank: the steady-state ranking is based on an eigenvector with eigenvalue 1.
• Vibrations: natural vibration modes are eigenvectors of the system matrix.

Similarity transformations

Two matrices are similar if

B = P^-1 A P

Similar matrices represent the same linear transformation in different bases. They have the same eigenvalues.

Key facts:

• Similar matrices share eigenvalues.
• trace(A) equals the sum of eigenvalues.
• det(A) equals the product of eigenvalues.

Common mistakes

• forgetting that an eigenvector must be non-zero;
• assuming all eigenvalues are positive;
• mixing up PDP^-1 and P^-1DP;
• assuming every matrix is diagonalizable.

Quick revision

• Solve Av = lambda v.
• Rearrange to (A - lambda I)v = 0.
• Find eigenvalues from det(A - lambda I) = 0.
• Then find eigenvectors from the null space of A - lambda I.
• Diagonalisation helps compute powers and understand stability.

Exam angle

If asked about eigenvalues and eigenvectors:

1. Define them using Av = lambda v.
2. Show the characteristic equation.
3. Solve one simple example.
4. Mention one application such as PCA or PageRank.