Mengxu Yuan | Searcharxiv

arXiv2026

Aggregate Bounds on the eigenvalues of the principal submatrices of a Hermitian matrix and majorization relations

We extend bounds, proved by R.C. Thompson in 1966, on the sum of the $j$-th largest eigenvalues of the $(n-1) \times (n-1)$ principal matrices of an $n \times n$ Hermitian matrix. Our bounds are stronger than just summing up Thompson's bounds. We achieve the extensions as a corollary of a more general result giving bounds on the zeros of the generalized derivatives of polynomials with real roots. We use the extended bounds to obtain majorization relationships between the eigenvalues of all $m \times m$ principal matrices of an $n \times n$ Hermitian matrix. These majorization relationships imply both a well-known majorization result by Schur and the well-known Szasz's inequalities.

math.RA↗

arXiv2026

Two absolutely bounded determinantal ratios

Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$ \det A \le A_{11} \cdots A_{nn}. $$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets $α_1, α_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[α_1 \cup α_2] \det A[α_1 \cap α_2] \le \det A[α_1] \det A[α_2], $$ where $A[α]$ denotes the principal submatrix determined by the indexes in $α$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.

math.RA↗