Mathematics.

linear transformations

Eigenspaces

Linear Algebra30 minDifficulty5 out of 10

You should know: eigenvalues and eigenvectors

Overview

The eigenspace of a matrix A for a given eigenvalue λ is the set of all eigenvectors with that eigenvalue, together with the zero vector — that is, the null space of A − λI. Every eigenspace is itself a subspace, and its dimension (called the geometric multiplicity of λ) tells you how many independent directions are scaled by that same factor λ. The geometric multiplicity is always at most the algebraic multiplicity (how many times λ is a repeated root of the characteristic polynomial); when they're equal for every eigenvalue, the matrix is diagonalizable.

Intuition

If an eigenvalue λ has more than one independent eigenvector, all of those eigenvectors — and every combination of them — form a whole subspace (a line, plane, or higher-dimensional flat) where the transformation acts as pure scaling by λ. Think of the identity matrix: every nonzero vector is an eigenvector with eigenvalue 1, so the entire space is one giant eigenspace. Contrast this with a matrix having a repeated eigenvalue but only one independent eigenvector for it — the eigenspace stays 'thin' (lower-dimensional than the multiplicity would allow), which is exactly the obstruction that prevents diagonalization.

Formal Definition

Definition

For a square matrix A and eigenvalue λ, the eigenspace E_λ is defined as:

Eλ={v∈Rn:Av=λv}=null⁡(A−λI)E_\lambda = \{v \in \mathbb{R}^n : Av = \lambda v\} = \operatorname{null}(A - \lambda I)
Eigenspace definition
1≤dim⁡(Eλ)≤ma(λ)1 \le \dim(E_\lambda) \le m_a(\lambda)
Geometric multiplicity ≤ algebraic multiplicity m_a
A diagonalizable  ⟺  dim⁡(Eλi)=ma(λi) for every eigenvalue λiA \text{ diagonalizable} \iff \dim(E_{\lambda_i}) = m_a(\lambda_i) \ \text{for every eigenvalue } \lambda_i
Diagonalizability criterion

Worked Examples

  1. Solve (A - 2I)v = 0.

    (A−2I)=(0000)(A - 2I) = \begin{pmatrix}0&0\\0&0\end{pmatrix}

  2. Every vector satisfies 0·v = 0, so the null space is all of ℝ².

    E2=R2,dim⁡(E2)=2E_2 = \mathbb{R}^2, \quad \dim(E_2) = 2

Answer: E_2 = ℝ² — the eigenspace is the entire plane, since A = 2I scales every vector by 2.

Practice Problems

Difficulty 5/10

Find the eigenspace of A = [[3,1],[0,2]] for λ = 3.

Difficulty 6/10

For A = [[4,0,0],[0,4,0],[0,0,5]], find dim(E_4) and dim(E_5).

Difficulty 6/10

In a vibration analysis, a structure's stiffness matrix has a repeated eigenvalue (natural frequency) with geometric multiplicity 2. What does this mean physically about the structure's mode shapes?

Quiz

The eigenspace E_λ of a matrix A is defined as:
Geometric multiplicity of an eigenvalue λ is always:
A matrix is diagonalizable exactly when:

Summary

  • The eigenspace E_λ = null(A - λI) is the subspace of all eigenvectors for eigenvalue λ (plus zero).
  • Geometric multiplicity (dim E_λ) is always ≤ algebraic multiplicity (root multiplicity in the characteristic polynomial).
  • A matrix is diagonalizable exactly when every eigenvalue's geometric and algebraic multiplicities match.

References

  1. WebsiteWikipedia — Eigenvalues and eigenvectors (Eigenspaces)