A Matrix Analogue of Schur-Siegel-Smyth Trace Problem
Let $\mathcal{S}$, $\mathcal{T}$, $\mathcal{C}$ be three sets of some special types of connected positive definite matrices. The \textit{Absolute Trace} of a square matrix $A$ is defined by $$ \frac{\text{Trace of matrix A}}{\text{Order of A}} $$ In this paper, we extend the notion of absolute trace by defining \textit{trace-$k$ measure} of $A$ $=$ mean value of the power sums of eigenvalues of $A$. And we obtain the greatest lower bound of this measure for any $n \times n $ matrix in $\mathcal{S} \cup \mathcal{T} \cup \mathcal{C}$. Using this we find some criteria which assure when an integer polynomial can not be a characteristic or minimal polynomial of an integer symmetric matrix. We also obtain here the smallest limit point of the set of all \textit{absolute trace-$k$ measure} of matrices in $\mathcal{S} \cup \mathcal{T} \cup \mathcal{C}$. Finally, we exhibit that the famous result of Smyth on density of absolute trace measure of totally positive integers is also true for the set of symmetric integer connected positive definite matrices.