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Eigenspace — Definition, Formula & Examples

An eigenspace is the set of all eigenvectors associated with a particular eigenvalue of a matrix, together with the zero vector. It forms a subspace of the vector space on which the matrix acts.

Given a square matrix AA and an eigenvalue λ\lambda, the eigenspace EλE_\lambda is the null space of the matrix (A−λI)(A - \lambda I). That is, Eλ={v∈Rn:(A−λI)v=0}E_\lambda = \{\mathbf{v} \in \mathbb{R}^n : (A - \lambda I)\mathbf{v} = \mathbf{0}\}. This set always includes the zero vector and is closed under addition and scalar multiplication, making it a subspace.

Key Formula

Eλ=ker⁡(A−λI)={v∈Rn:(A−λI)v=0}E_\lambda = \ker(A - \lambda I) = \{\mathbf{v} \in \mathbb{R}^n : (A - \lambda I)\mathbf{v} = \mathbf{0}\}
Where:
  • AA = An $n \times n$ square matrix
  • λ\lambda = An eigenvalue of $A$
  • II = The $n \times n$ identity matrix
  • v\mathbf{v} = A vector in $\mathbb{R}^n$

How It Works

To find the eigenspace for a given eigenvalue λ\lambda, you form the matrix A−λIA - \lambda I and then solve the homogeneous system (A−λI)v=0(A - \lambda I)\mathbf{v} = \mathbf{0}. Row-reduce A−λIA - \lambda I to find the free variables, then express the general solution as a linear combination of basis vectors. Those basis vectors span the eigenspace. The dimension of the eigenspace is called the geometric multiplicity of λ\lambda.

Worked Example

Problem: Find the eigenspace of the matrix A=(4213)A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} corresponding to the eigenvalue λ=2\lambda = 2.
Step 1: Compute A−λIA - \lambda I by subtracting 22 from each diagonal entry.
A−2I=(4−2213−2)=(2211)A - 2I = \begin{pmatrix} 4-2 & 2 \\ 1 & 3-2 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}
Step 2: Row-reduce to solve (A−2I)v=0(A - 2I)\mathbf{v} = \mathbf{0}. Replace R1R_1 with R1−2R2R_1 - 2R_2.
(2211)→(0011)\begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix} \to \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}
Step 3: The system gives v1+v2=0v_1 + v_2 = 0, so v1=−v2v_1 = -v_2. Let v2=tv_2 = t.
v=t(−11),t∈R\mathbf{v} = t\begin{pmatrix} -1 \\ 1 \end{pmatrix}, \quad t \in \mathbb{R}
Answer: The eigenspace E2E_2 is span{(−11)}\text{span}\left\{\begin{pmatrix} -1 \\ 1 \end{pmatrix}\right\}, a one-dimensional subspace of R2\mathbb{R}^2.

Why It Matters

Eigenspaces determine whether a matrix is diagonalizable: a matrix can be diagonalized exactly when the eigenspaces together span the full vector space. This arises directly in solving systems of differential equations, principal component analysis, and quantum mechanics, where the structure of eigenspaces dictates the behavior of the system.

Common Mistakes

Mistake: Forgetting that the eigenspace includes the zero vector and calling it just the set of eigenvectors.
Correction: The eigenspace is a subspace, so it must contain the zero vector by definition. The zero vector itself is not an eigenvector, but it is an element of every eigenspace.