Eigenvectors

Level 1 - Math I (Physics) topic page in Linear Maps.

Principle

An eigenvector is a non-zero vector whose direction is preserved by a linear map. The vector may be stretched, shrunk, reversed, or unchanged, but it stays on the same line.

Eigenvector equation

\[A\mathbf v=\lambda\mathbf v\]

In physics, eigenvectors often describe independent modes: principal axes, polarisation states, or directions that respond without mixing.

Notation

\(\mathbf v\)

non-zero eigenvector

\(\lambda\)

eigenvalue belonging to the eigenvector

\(A-\lambda I\)

matrix used to solve for eigenvectors

\(\mathbf 0\)

zero vector

\(c\mathbf v\)

non-zero scalar multiple of an eigenvector

If \(\\mathbf v\) is an eigenvector, every non-zero multiple of \(\\mathbf v\) is also an eigenvector with the same eigenvalue.

Method

Step 1: Start from a known eigenvalue

Eigenvectors are found after an eigenvalue has been identified.

Step 2: Solve a homogeneous system

Substitute \(\\lambda\) into \(A-\\lambda I\) and solve

Eigenvector system

\[(A-\lambda I)\mathbf v=\mathbf 0\]

Step 3: Exclude the zero vector

The zero vector always solves the homogeneous system, but it is not an eigenvector.

Rules

Scalar multiple rule

\[A(c\mathbf v)=cA\mathbf v=c\lambda\mathbf v=\lambda(c\mathbf v)\]

Non-zero condition

\[\mathbf v\ne\mathbf 0\]

Eigenvectors for distinct eigenvalues are linearly independent in standard finite-dimensional settings.

Examples

Question

Find an eigenvector for

\[A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\]

with

\[\lambda=2\]

Answer

Solve

\[(A-2I)\mathbf v=\mathbf0\]

This gives

\[y=0\]

so one eigenvector is

\[(1,0)\]

Checks

Do not list the zero vector as an eigenvector.
Eigenvectors belong to a particular eigenvalue.
Any non-zero scalar multiple represents the same eigenvector direction.
Solve \((A-\\lambda I)\\mathbf v=\\mathbf0\), not \(A\\mathbf v=\\mathbf0\) unless \(\\lambda=0\).

Eigenvalues

Eigenspaces