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Double-centering a finite positive-semidefinite Gram matrix preserves positive semidefiniteness

Statement

Let $n \geq 1$ and let $K \in R^{n \times n}$ be symmetric and satisfy $a^{T} K a \geq 0$ for every $a \in R^{n}$ . With $H_{n} = I_{n} - n^{- 1} 1_{n} 1_{n}^{T}$ , the centered matrix $K_{c} = H_{n} K H_{n}$ is symmetric and positive semidefinite. Consequently it has an orthonormal eigenbasis and all its eigenvalues are nonnegative.

Statement references

Hastie, Tibshirani, and Friedman, ESL II, §14.5.4, printed pp. 547–550, and §18.5.2, printed pp. 670–672

Cameron, Linear and Nonlinear Dimensionality Reduction, §2.2, PDF pp. 3–4

Facts & Assumptions

Given: A symmetric positive-semidefinite matrix $K \in R^{n \times n}$ and its centering matrix $H_{n}$ .

[F1]

Kernel PCA takes a symmetric matrix $K$ with $a^{T} K a \geq 0$ for every $a$ and centers it as $K_{c} = H_{n} K H_{n}$ (Kernel principal component analysis for a finite positive-semidefinite Gram matrix).

[F2]

The finite centering matrix is symmetric (The finite centering matrix is the orthogonal projection onto the zero-sum subspace).

[F3]

The standard inner product on $R^{n}$ is $⟨ a, b ⟩ = \sum_{i < n} a_{i} b_{i}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F4]

In an orthonormal basis, a real matrix is self-adjoint exactly when it is symmetric (In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose).

[F5]

Every finite-dimensional real self-adjoint endomorphism has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

Proof

1.1F1F2algebra

Since $K^{T} = K$ and $H_{n}^{T} = H_{n}$ by [F1] and [F2], $K_{c}^{T} = (H_{n} K H_{n})^{T} = H_{n} K H_{n} = K_{c}$ .

1.2givenF1F2algebra

For every $a \in R^{n}$ , $a^{T} K_{c} a = (H_{n} a)^{T} K (H_{n} a) \geq 0$ by the input hypothesis on $K$ . Thus $K_{c}$ is positive semidefinite.

2.1step 1.1step 1.2F3F4F5∎

The standard coordinate basis is orthonormal for [F3], so [F4] makes the symmetric matrix $K_{c}$ self-adjoint and [F5] gives an orthonormal eigenbasis. If $u$ is a unit eigenvector with eigenvalue $λ$ , then $λ = u^{T} K_{c} u \geq 0$ by step 1.2. Hence every eigenvalue is nonnegative, which makes the stated kernel-PCA eigenprocedure well-defined.

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Kernel principal component analysis for a finite positive-semidefinite Gram matrix.

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Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)