The expected characteristic and permanental polynomials of the random Gram matrix

TL;DR

This paper derives recursive formulas and generating functions to efficiently compute the expected characteristic and permanental polynomials of random Gram matrices formed from independent columns, linking their spectral properties to combinatorial quantities.

Contribution

It introduces new theorems relating traces, characteristic coefficients, determinants, and permanents of random Gram matrices, enabling faster computation and deeper analysis.

Findings

Derived recursive formulas for expected determinants and permanents

Established generating functions linking traces and polynomial coefficients

Provided methods for efficient numerical computation of expectations

Abstract

A t by n random matrix A is formed by sampling n independent random column vectors, each containing t components. The random Gram matrix of size n, G_n, contains the dot products between all pairs of column vectors in the randomly generated matrix A; that is, G_n = transpose(A) A. The matrix G_n has characteristic roots coinciding with the singular values of A. Furthermore, the sequences det(G_i) and per(G_i) (for i = 0, 1, ..., 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(per(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.