The Factorization of a Sum of Matrices and the Multivariate Cumulants of a Set of Quadratic Expressions

A factorization of I − B1 − B2 − … − Bm is given in the form Πn=1∞(I − Pn) where each Pn is a product of B's. Up to terms of degree n there are ∏r=1n1r∑d|μ1rmd! possible factorizations. The P's are called prime words since they have some of the properties of prime numbers. A theorem concerning prime words, resembling the ordinary prime number theorem, is conjectured and partly proved. A simple method is given for writing down, in terms of the traces of the “prime words” and their “powers,” the cumulants of the joint distribution of a set of quadratic expressions in a vector with a multinormal distributions.