The expected characteristic and permanental polynomials of the random Gram matrix

• Main Authors: Martin, Jacob G, Canfield, E. Rodney
• Format: Journal Article
• Language: English
• Published: 09-09-2013
• Subjects: Mathematics - Combinatorics; Mathematics - Probability
• DOI: 10.48550/arxiv.1309.2599

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.