Math 289A: Lecture 8

Let X_{MN} be a random rectangular matrix, i.e. a random variable taking values in \mathbb{C}^{M\times N} viewed as a real vector space with scalar product

\langle T,X \rangle = \Re\mathrm{Tr}\, T^*X = \frac{1}{2}\mathrm{Tr}(T^*X+X^*T).

The characteristic function of X_{MN} is the function on 2MN-dimensional Euclidean space \mathbb{C}^{M \times N} defined by the expectation

\varphi_{X_{MN}}(T) = \mathbb{E}\left[e^{i \langle T,X_{MN}\rangle}\right],

which is the Fourier transform

\varphi_{X_{MN}}(T) = \int\limits_{\mathbb{C}^{M\times N}}e^{ \frac{i}{2}\mathrm{Tr}(T^*X+X^*T)}\mu_{MN}(\mathrm{d}X)

of the distribution of \mu_{MN} of X_{MN} in \mathbb{C}^{M \times N} \simeq \mathbb{R}^{2MN}.

The product group \mathbb{U}_{MN} = \mathbb{U}_M \times \mathbb{U}_N acts on \mathbb{C}^{M \times N} by bi-conjugation: we have a group homomorphism \Phi from \mathbb{U}_{MN} into the orthogonal group of the Euclidean space \mathbb{C}^{M \times N} defined by

\Phi(U,V)X = UXV^*, \quad (U,V) \in \mathbb{U}_{MN},\ X \in \mathbb{C}^{M \times N}.

Problem 8.1. Check that \Phi really is a group homomorphism, and that it really does preserve the Euclidean scalar product on \mathbb{C}^{M \times N}.

Let us assume M \geq N, so that the product T^*X is an N \times N square matrix which fits inside an M \times N rectangle, rather than containing it. The action of \mathbb{U}_{MN} on \mathbb{C}^{M \times N} is the subject of the rectangular analogue of the spectral theorem. The rectangular spectral theorem is called the singular value decomposition, and it says that that the biconjugation orbit of any given rectangular matrix X \in \mathbb{C}^{M \times N} contains an M \times N real diagonal matrix B which is unique up to signed permutations of its diagonal elements. In particular, there is one and only one diagonal matrix B in the biconjugation orbit of X whose entries satisfy

b_1 \geq \dots \geq b_N \geq 0,

and these numbers are referred to as the singular values of X.

Definition 8.1. We say that a random rectangular matrix X_{MN} is unitarily bi-invariant if its characteristic function is invariant under unitary bi-conjugation,

\varphi_{X_{MN}}(T) = \varphi_{X_{MN}}(UTV^*), \quad T \in \mathbb{C}^{M \times N},\ (U,V) \in \mathbb{U}_{MN}.

Problem 8.2. We proved that a unitarily invariant random Hermitian matrix is uniquely determined by the joint law of its diagonal matrix elements. State and prove the analogue of this theorem for unitarily bi-invariant random rectangular matrices.

Theorem 8.1. A random rectangular matrix X_{MN} is unitarily bi-invariant if and only if its distribution is stable under unitary bi-conjugation.

Proof: By cyclic invariance of the trace, we have

\varphi_{UX_{MN}V^*}(T)= \varphi_{X_{MN}}(U^*TV).

-QED

Corollary 8.1. A random rectangular matrix X_{MN} is unitarily bi-invariant if and only if it has the same law as a product of the form U_MB_{MN}V_N^*, where (U_M,V_N) is a uniformly random sample from \mathbb{U}_{MN}, and B_{MN} is an N-dimensional real random vector viewed as an M \times N diagonal matrix.

Equivalently, Corollary 8.1 says that a unitarily bi-invariant random rectangular matrix X_{MN} is uniformly random conditional on the law of its singular values. In terms of characteristic functions, we can formulate this as follows.

Definition 8.2. The quantum Bessel kernel

K_{MN} \colon \mathbb{R}^N \times \mathbb{R}^N \longrightarrow \mathbb{C}

is defined by

K_{MN}(A,B) = \mathbf{E}[e^{i\langle A,U_MBV_N^*\rangle}]=\int\limits_{\mathbb{U}_{MN}} e^{\frac{i}{2}\mathrm{Tr}\,(A^*UBV^*+VB^*U^*A)} \mathrm{d}(U,V),

where A,B \in \mathbb{R}^N are viewed as M \times N diagonal matrices.

The quantum Bessel kernel K_{MN} is the characteristic function of a uniformly random M \times N rectangular matrix with prescribed deterministic singular. This is the rectangular counterpart of the quantum Fourier kernel K_N, which is the characteristic function of a uniformly random Hermitian matrix with prescribed singular values. Note that, even when M=N, the kernels K_{NN} and K_N are not the same thing. In particular, look at the case M=N=1. The quantum Fourier kernel reverts to the classical Fourier kernel on the line,

K_1(a,b) = e^{iab},

which is tautologically the same thing as the characteristic function of a real random variable whose distribution is a delta measure, i.e. the Fourier transform of a point mass. However, the quantum Bessel kernel at M=N=1 is something else, namely

K_{11}(a,b) = \int\limits_{\mathbb{U}_{11}} e^{\frac{i}{2}(aub\bar{v} + vb\bar{u}a)}\mathrm{d}(u,v) = \int\limits_{\mathbb{U}_1} e^{\frac{i}{2}( abu + ab\bar{u})}\mathrm{d}u.\

This is a nontrivial transform, namely the characteristic function of a uniformly random point on a circle of radius |b|, the power series expansion of which is

K_{11}(a,b) = 1 + \sum\limits_{d=1}^\infty \frac{i^{2d}}{2^{2d}d!d!}(ab)^{2d}=1 + \sum\limits_{d=1}^\infty \frac{\left(\frac{1}{2}iab\right)^{2d}}{d!d!}.

Stop a random person on the street and they will almost surely recognize this as the Bessel kernel J_0(ab) which defines the Hankel transform of a random vector in \mathbb{R}^2 with radially symmetric distribution.

Definition 8.3. Given an N-dimensional real random vector B_N and any integer M\geq N, the Mth quantum Hankel transform of B_N is defined to be the expectation

\psi_{MN}(A) = \mathbf{E}K_{MN}(A,B_N), \quad A \in \mathbb{R}^N.

The random matrix calculation we have done above can be stated as follows.

Theorem 8.2. The classical characteristic function of a unitarily bi-invariant random rectangular matrix X_{MN} is equal to the quantum Hankel transform of its singular values B_{MN}.

Published by Jonathan Novak

Mathematician at UC San Diego.

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