Eigenvalues

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

Principle

An eigenvalue is a scale factor for a direction that a linear map does not turn. If \(A\\mathbf v\) points along the same line as \(\\mathbf v\), then the scale factor is an eigenvalue.

Eigenvalue equation

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

Eigenvalues identify natural rates, growth factors, decay factors, and squared frequencies in many linear physics models.

Notation

\(A\)

square matrix representing a linear map

\(\lambda\)

eigenvalue

\(\mathbf v\)

non-zero eigenvector

\(I\)

identity matrix of the same size as A

\(\det(A-\lambda I)\)

characteristic polynomial

Eigenvalues are defined only for square matrices, because input and output vectors must live in the same space.

Method

Step 1: Rearrange the eigenvalue equation

Move all terms to one side.

Homogeneous form

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

Step 2: Require a non-zero solution

There is a non-zero solution only when \(A-\\lambda I\) is singular.

Characteristic equation

\[\det(A-\lambda I)=0\]

Step 3: Solve for eigenvalues

Expand the determinant and solve the resulting polynomial equation.

Rules

Characteristic polynomial

\[p(\lambda)=\det(A-\lambda I)\]

Eigenvalue condition

\[p(\lambda)=0\]

Two by two case

\[(a-\lambda)(d-\lambda)-bc=0\]

Examples

Question

Find the eigenvalues of

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

Answer

The characteristic equation is

\[(2-\lambda)(5-\lambda)=0\]

Therefore the eigenvalues are

\[\lambda=2\]

and

\[\lambda=5\]

Checks

Eigenvalues require a square matrix.
The eigenvector must be non-zero.
Solve \(\\det(A-\\lambda I)=0\), not \(\\det A=0\) unless \(\\lambda=0\).
Repeated eigenvalues need extra care when finding eigenvectors.

Special Matrices

Eigenvectors