Abstract
A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.
Highlights
Let w be a t-tall vector whose components wi are random variables w1 w = w2 ... wtsample n independent vectors w(1), . . . , w(n); creating a t × n matrix A
We have introduced the notion of a random Gram matrix, and provided theory enabling the efficient computation of the expected determinant and expected permanent of it
The expected coefficients of the characteristic and permanental polynomials have been studied, with some numerical experiments checking on the theory
Summary
Let w be a t-tall vector whose components wi are random variables (not necessarily independent) w1. The paper’s main result, which treats the particular case of the Gaussian unitary ensemble, is derived from an expression of an arbitrary n × n permanent as a 2n-dimensional contour integral (see Lemma 2.1). The integrand in the latter formula is the exponential of the trace of a certain tensor product. We present combinatorial theory and an efficient algorithm for calculating E(det(Gn)) and E(perm(Gn)), which are factors comprising the coefficients of the expected characteristic and expected permanental polynomials of Gn. The computation of the determinant is equivalent to matrix multiplication and is contained in the complexity class P (see Chapter 16 of [5]). The rest of the paper is organized as follows: Section 2 is a statement of results, Section 3 contains all proofs, Section 4 reports on some numerical experiments, Section 5 points out a connection to prior work involving the cycle index polynomial of the symmetric group, and Section 6 presents summary and conclusions
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