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A Hermitian directional spectral measure is a probability law

Statement

For every positive integer $d$ and every Hermitian positive semidefinite $A \in C^{d \times d}$ , its empirical spectral distribution $μ_{A}$ is a Borel probability measure supported on $[0, \infty)$ . For every deterministic unit vector $x \in C^{d}$ , its directional spectral measure $μ_{A}^{x}$ is also a Borel probability measure supported on $[0, \infty)$ .

Statement references

Bai, Miao, and Pan, §1, printed pp. 7–8 (full paper PDF).

Bai and Silverstein, Appendix B.2, printed pp. 514–523 (full book PDF).

Directional and empirical spectral-measure definitions: definition of the directional spectral measure.

Facts & Assumptions

Given: A positive integer $d$ , a Hermitian positive semidefinite matrix $A \in C^{d \times d}$ and, for the directional measure, a deterministic vector $x \in C^{d}$ with $∥ x ∥_{2} = 1$ .

[F1]

A finite-dimensional Hermitian matrix has an orthonormal eigenbasis; its eigenvalues are real, and positive semidefiniteness makes them nonnegative (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F2]

The empirical and directional measures are the finite Dirac sums stated in Directional spectral measure of a Hermitian matrix.

[F3]

A Borel probability measure is countably additive and has total mass one (Probability measures and probability spaces).

[F4]

The Dirac set function $δ_{λ}$ defined in The Dirac set function at a point is a Borel probability measure by A Dirac set function is a probability measure.

Proof

Given: $A$ , $x$ , and the definitions in the Statement.

1.1givenF1F2F3F4algebra

By [F1], write $A = \sum_{j = 1}^{r} λ_{j} P_{j}$ over its distinct eigenvalues, where the $P_{j}$ are mutually orthogonal projections, $\sum_{j} P_{j} = I$ , and $λ_{j} \geq 0$ . Taking traces gives $\sum_{j} rank (P_{j}) = d$ . Thus the nonnegative weights $rank (P_{j}) / d$ in [F2] sum to one; by [F3] and [F4], $μ_{A}$ is a Borel probability measure supported on $[0, \infty)$ .

2.1givenF1F2F3F4step 1.1algebra∎

For the unit vector $x$ , the directional weights in [F2] are $∥ P_{j} x ∥_{2}^{2} \geq 0$ , and orthogonality gives $\sum_{j} ∥ P_{j} x ∥_{2}^{2} = ∥ \sum_{j} P_{j} x ∥_{2}^{2} = ∥ x ∥_{2}^{2} = 1$ . By [F3] and [F4], the finite Dirac sum is a Borel probability measure; its support is contained in the nonnegative spectrum of $A$ by [F1].

Source notes

Bai, Miao, and Pan, equation (1.8), printed p. 8 (full paper PDF).

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Sources

Z. D. Bai, B. Q. Miao, and G. M. Pan, On Asymptotics of Eigenvectors of Large Sample Covariance Matrix (standard reference, not scraped)
Z. D. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. (standard reference, not scraped)