Statistics and characterization of matrices by determinant and trace

Abstract

Answering a question of Erdös, Komlós proved in 1968 that almost all n×n Bernoulli matrices are nonsingular as n→∞. In this paper, we offer a new perspective on the question of Erdös by studying n×n matrices with prime number entries in an almost all sense. Precisely, it is shown that, as x→∞, the probability of randomly choosing a nonsingular n×n matrix among all n×n matrices with prime number entries that are ≤x is 1. If A is a unitary matrix, then it is well known that |detA|=1. However, the converse is far from being true. As a remedy of this defect, we search for necessary and sufficient conditions for being a unitary matrix by teaming up determinant with trace. In this way, we are led to simple characterizations of unitary matrices in the set of normal matrices. The question of which nonsingular commuting complex matrices with real eigenvalues have the same characteristic polynomial is formulated via determinant and trace conditions. Finally, through a study of eigenvectors, we obtain new characterizations of Hermitian and normal matrices. Our approach to proving these results benefits from a modular interpretation of nonsingularity and the spectral theorem for normal operators together with equality cases of classical inequalities such as the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality.