Eigenspaces

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

Principle

An eigenspace is the set of all eigenvectors for one eigenvalue, together with the zero vector. It is the solution space of a homogeneous linear system.

Eigenspace

\[E_{\lambda}=\ker(A-\lambda I)\]

Eigenspaces organise eigenvectors into vector spaces. A whole line, plane, or higher-dimensional subspace can share the same scaling factor.

Notation

\(E_{\lambda}\)

eigenspace belonging to eigenvalue lambda

\(\ker(A-\lambda I)\)

null space of A minus lambda I

\(\dim E_{\lambda}\)

dimension of the eigenspace

\(\operatorname{span}\{\mathbf v_1,\mathbf v_2\}\)

all linear combinations of the listed vectors

The zero vector is included so that the eigenspace is a subspace, even though the zero vector is not an eigenvector.

Method

Step 1: Choose one eigenvalue

Work with one \(\\lambda\) at a time.

Step 2: Solve the null-space equation

Null-space equation

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

Step 3: Write the solution as a span

Use free variables to express every solution as a linear combination of basis vectors for the eigenspace.

Rules

Eigenspace definition

\[E_{\lambda}=\{\mathbf v:(A-\lambda I)\mathbf v=\mathbf0\}\]

Dimension

\[\dim E_{\lambda}=\text{number of independent eigenvector directions}\]

For distinct eigenvalues, their eigenspaces meet only at the zero vector in common finite-dimensional cases.

Examples

Question

Find the eigenspace for

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

with

\[\lambda=5\]

Answer

Solve

\[(A-5I)\mathbf v=\mathbf0\]

This gives

\[x=0\]

and \(y\) free, so

\[E_5=\operatorname{span}\{(0,1)\}\]

Checks

Work with one eigenvalue at a time.
Include the zero vector in the eigenspace, but do not call it an eigenvector.
Write eigenspaces as spans when possible.
The dimension of an eigenspace counts independent eigenvector directions.

Eigenvectors

Diagonalisation