Home Statistical Dictionary Eigendecomposition

Eigendecomposition

The factorization of a matrix into its eigenvectors and eigenvalues, revealing the directions the matrix only stretches (never rotates) and by how much -- the general decomposition that, applied specifically to a symmetric matrix, is called the spectral decomposition and underlies PCA.

Linear Algebra for StatisticsAdvancedFree

In Plain English

Eigendecomposition finds the special directions a matrix treats simply, directions it only stretches or shrinks, without rotating or bending them at all, and reports exactly how much stretching happens along each one. When the matrix in question is symmetric, like a covariance matrix, this decomposition becomes especially well-behaved (see Spectral Decomposition), and the resulting directions and stretch factors are exactly what principal component analysis reports as its components and their explained variances.

Definition

The eigendecomposition of a diagonalizable square matrix $A$ expresses it as $A = VΛ V^{- 1}$ , where $V$ is the matrix whose columns are $A$ 's eigenvectors and $Λ$ is a diagonal matrix of the corresponding eigenvalues, satisfying $A v_{i} = λ_{i} v_{i}$ for each eigenvector-eigenvalue pair. Not every square matrix is diagonalizable (some lack a full set of $n$ linearly independent eigenvectors), but every real symmetric matrix is always diagonalizable, with real eigenvalues and an orthogonal eigenvector matrix, this special, always-well-behaved case is specifically called the spectral decomposition, and is the version that applies directly to covariance matrices and underlies PCA.

Formula

A = VΛ V^{- 1}

Notation

V

—matrix of eigenvectors (columns)

Λ

—diagonal matrix of eigenvalues

λ_{i}, v_{i}

—the i-th eigenvalue and its corresponding eigenvector

Properties

An eigenvector is a direction a matrix leaves unrotated, only scaling it by the corresponding eigenvalue -- this is the geometric heart of what eigendecomposition reveals, the matrix's action, however complicated it looks in the original coordinate system, is just simple scaling once viewed along its own eigenvector directions.
Not every square matrix has a full eigendecomposition, a matrix that lacks a complete set of $n$ linearly independent eigenvectors (a 'defective' matrix) cannot be diagonalized this way and instead requires the more general Jordan normal form -- but this complication never arises for the symmetric matrices (covariance, correlation, X'X) that dominate applied statistics.
The determinant of a matrix equals the product of its eigenvalues and its trace equals their sum, two of the most commonly used matrix summaries are both direct, simple functions of the same eigenvalues an eigendecomposition reveals, which is part of why eigendecomposition is considered such a foundational tool rather than a narrow specialty technique.

At a Glance

FactorizationA = V (Lambda) V^-1, V's columns are eigenvectors, Lambda is diagonal eigenvalues

Eigenvector propertyA v_i = lambda_i v_i -- the matrix only scales these directions

Not every matrixis diagonalizable this way -- requires n independent eigenvectors

Always works forsymmetric matrices -- this special case is the spectral decomposition

Last updated September 8, 2026← Back to the Dictionary