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A repeated leading PCA eigenvalue identifies a subspace, not a vector

Statement

Let $Σ$ be a covariance matrix in the PCA definition of Covariance and correlation principal component analysis, with ordered eigenvalues $λ_{1} \geq λ_{2} \geq \dots$ . If $λ_{1} = λ_{2}$ , then the leading eigenspace $E_{1} = \ker (Σ - λ_{1} I)$ is a uniquely specified subspace of dimension at least two. Every unit vector in $E_{1}$ is a leading principal direction, so the covariance matrix determines this subspace but does not single out a unit vector in it.

Facts & Assumptions

Given: A real symmetric covariance matrix $Σ \in R^{p \times p}$ with $p \geq 2$ and $λ_{1} = λ_{2}$ .

[F1]

Covariance and correlation principal component analysis: covariance-PCA uses the ordered eigenspaces of the symmetric covariance matrix; a component direction is a unit vector in its eigenspace.

[F2]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: $Σ$ has an orthonormal eigenbasis, so the eigenspace dimension of an eigenvalue is its multiplicity in the ordered list.

[F3]

The leading PCA eigenspace maximizes projected variance: the unit leading directions are exactly the unit vectors in the leading eigenspace.

Proof

technique · direct

1.1F1F2givenalgebra

In the orthonormal eigenbasis supplied by [F2], the equality $λ_{1} = λ_{2}$ gives at least two linearly independent eigenvectors with eigenvalue $λ_{1}$ . The full eigenspace $E_{1} = \ker (Σ - λ_{1} I)$ contains these vectors, is specified by $Σ$ itself, and therefore has dimension at least two.

2.1F1F3step 1.1algebra∎

Every unit vector in $E_{1}$ is a leading principal direction by [F3]. Because a space of dimension at least two contains noncollinear unit vectors, there is no single unit vector selected by $Σ$ alone. The canonical object is the whole leading eigenspace $E_{1}$ .

Depends on

Used by

Tied PCA eigenvalues and sign ambiguity Example

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Online Bayesian Data Analysis, Lecture Notes 8: Learning Signal Representations (11 April 2016) (standard reference, not scraped)
Soumendu Sundar Mukherjee, Multivariate Analysis, Lecture 8: Principal Component Analysis (standard reference, not scraped)