Math 289A: Lecture 4

We have defined the characteristic function of a random Hermitian matrix X according to the usual recipe for random variables taking values in a Euclidean space: it is the expectation

\varphi_{X}(T) = \mathbf{E}\left[e^{i\langle T,X \rangle}\right],

where the argument T ranges over the real vector space

\mathbb{H}_N = \{X \in \mathbb{C}^{N \times N} \colon X^*=X\},

which carries the Hilbert-Schmidt scalar product

\langle T,X \rangle = \mathrm{Tr}\, TX.

Equivalently,

\varphi_{X}(T) = \int\limits_{\mathbb{H}_N} e^{i \mathrm{Tr}\, TX}\mu_X(\mathrm{d}X),

where \mu_X is the distribution of X in \mathbb{H}. With this definition in hand, we can consider special classes of random Hermitian matrices carved out by clauses of the form

“A random Hermitian matrix X is said to be (descriptor) if its characteristic function satisfies (property).”

A natural construction of this form arises from the Spectral Theorem. The set

\mathbb{U}_N=\{U \in \mathbb{C}^{N \times N} \colon UU^*=UU^*=I\}

of unitary matrices is a group under matrix multiplication, called the unitary group of rank N. The group \mathbb{U}_N acts on the Euclidean space \mathbb{H}_N by conjugation, which is a group homomorphism

\Phi \colon \mathbb{U}_N \longrightarrow \mathrm{Aut}(\mathbb{H}_N,\langle \cdot,\cdot\rangle)

from \mathbb{U} into the orthogonal group of the Euclidean space \mathbb{H} defined by

\Phi(U)T = UTU^*, \quad U \in \mathbb{U}_N,\ T \in \mathbb{H}_N.

Problem 4.1. Prove that \Phi really is a group homomorphism from \mathbb{U}_N into orthogonal transformations of Euclidean space \mathbb{H}_N.

One formulation of the Spectral Theorem says: the conjugation orbit

\mathcal{O}_X = \{UXU^* \colon U \in \mathbb{U}_N\}

of any Hermitian matrix X \in \mathbb{H}_N contains a diagonal matrix A, which is unique up to simultaneous permutations of its rows and columns.

Definition 4.1. A random Hermitian matrix X is said to be unitarily invariant if its characteristic function \varphi_{X} satisfies

\varphi_{X}(UTU^*) = \varphi_{X}(T)

for all T \in \mathbb{H}_N and all U \in\mathbb{U}_N. That is, X is unitarily invariant if its characteristic function \varphi_{X} is constant on each \mathbb{U}_N-orbit of \mathbb{H}_N.

Unitary invariance is a basic symmetry with many ramifications. First of all, it leads to a massive reduction in complexity by allowing us to view the characteristic function of X as a function of A \in \mathbb{R}^N rather than a function of T \in \mathbb{H}_N. Indeed, for any arbitrary Hermitian T, the Spectral Theorem says that UTU^*=A for some unitary matrix U \in \mathbb{U}_N and some diagonal matrix A \in \mathbb{H}_N. Unitary invariance of the characteristic function then gives

\varphi_{X}(T) = \varphi_{X}(UTU^*) = \varphi_{X}(A).

The diagonal Hermitian matrix

A = \begin{bmatrix} a_1 & {} & {} \\ {} & \ddots & {} \\ {} & {} & a_N\end{bmatrix}

can be viewed as a vector A =(a_1,\dots,a_N) \in \mathbb{R}^N, so that the characteristic function of a unitarily invariant random Hermitian matrix X can be viewed as a continuous function

\varphi_{X} \colon \mathbb{R}^N \longrightarrow \mathbf{D}

on N-dimensional Euclidean space taking values in the closed unit disc \mathbf{D} \subset \mathbb{C}.

Problem 4.2. Prove that the characteristic function of a unitarily invariant random Hermitian matrix X is a symmetric function on \mathbb{R}^N.

Now we discuss the probabilistic ramifications of unitary invariance, i.e. what Definition 4.2 implies about the distribution of X.

Theorem 4.1. The distribution of a unitarily invariant random Hermitian matrix X is completely determined by the joint distribution of its diagonal matrix elements X_{11},\dots,X_{NN}, which are exchangeable real random variables.

Proof: For A=\mathrm{diag}(a_1,\dots,a_N) real diagonal, we have

\mathrm{Tr}\, AX = a_1X_{11} + \dots + a_NX_{NN}.

Consequently, we have

\varphi_{X}(A) = \mathbb{E}[e^{i\mathrm{Tr}\, AX}] = \mathbb{E}[e^{i(a_1X_{11} + \dots + a_NX_{NN})}],

which is exactly the characteristic function of the real random vector

D = (X_{11},\dots,X_{NN}).

Thus, the characteristic function \varphi_X(A)=\varphi_D(A). The exchangeability of X_{11},\dots,X_{11} follows from Problem 4.1.

-QED

It is important to note that exchangeable random variables are identically distributed, but not necessarily independent.

Problem 4.3. Prove that the diagonal matrix elements of a unitarily invariant random Hermitian matrix X are independent if and only if the distribution \mu_{X} is a Gaussian measure on \mathbb{H}_N.

Now we characterize unitary invariance as a feature of the distribution of a random Hermitian matrix.

Theorem 4.2. A random Hermitian matrix X is unitarily invariant if and only if X and UX_NU^* have the same distribution for every U \in \mathbb{U}_N.

Proof: Note that for any random Hermitian matrix X and any deterministic unitary matrix U \in \mathbb{U}_N we have

\varphi_{UXU^*}(T) =\varphi_{X}(U^*TU),\quad T \in \mathbb{H}_N,

by cyclic invariance of the trace. If X is unitarily invariant, this becomes

\varphi_{UXU^*}(T) =\varphi_{X}(T), \quad T \in \mathbb{H}_N,

which is equivalent to equidistribution of X and UX_NU^*. Conversely, if UXU^* and X have the same distribution then their characteristic functions are equal.

-QED

Theorem 4.2 is quite important for the following reason. Define a random linear operator acting on \mathbb{C}^N by

L_Xv = Xv, \quad v \in \mathbb{C}^N.

The matrix of L_{X} in the standard basis e_1,\dots,e_N of \mathbb{C}^N is X. The matrix of L_{X} in some other orthonormal basis f_1,\dots,f_N is the UXU^*, where U is the matrix of the linear transformation defined by e_i \mapsto f_i. Unitary invariance says that UXU^* has the same distribution as X – the random matrix of the random operator L_{X} has the same law for any choice of orthonormal basis in the Hilbert space \mathbb{C}^N. In this sense, unitarily invariant random Hermitian matrices are precisely those random matrices which correspond to canonically defined random Hermitian operators on finite-dimensional Hilbert space.

Published by Jonathan Novak

Mathematician at UC San Diego.

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