arXiv subjects

Hristo Sendov

Publications and source records attributed to Hristo Sendov.

7 recordsLinked to original sources

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 $\alpha_1, \alpha_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[\alpha_1 \cup \alpha_2] \det A[\alpha_1 \cap \alpha_2] \le \det A[\alpha_1] \det A[\alpha_2], $$ where $A[\alpha]$ denotes the principal submatrix determined by the indexes in $\alpha$. 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↗

arXiv2026

A Bounded Determinantal Ratio for Positive Definite Matrices

Bounded ratios between products of minors of a positive definite matrix have a long history. Starting with Hadamard's inequality which bounds from above the determinant by the product of its diagonal entries and progressing through its generalizations the Fischer's inequality and the Koteljanskii's inequality. Finding new bounded determinantal ratios is a difficult task and finding their supremum is even more difficult. In 2008, Hall and Johnson showed that the ratio $$ \frac{\det A[\{1,2,4\}] \det A[\{1,3,4\}] \det A[\{2,3\}] \det A[\{1\}] \det A[\{4\}]}{\det A[\{1,2\}] \det A[ \{1,3\}] \det A[\{1,4\}] \det A[\{2,4\}] \det A[\{3,4\}]} $$ is bounded above by $4$. They conjectured that the supremum of the ratio, over all $4 \times 4$ positive definite matrices $A$, is $27/16$. Here, $A[\alpha]$ denotes the principal minor of $A$ corresponding to the rows and columns indexed by $\alpha \subseteq \{1,2,3,4\}$. In this paper we confirm that the supremum of the ratio is $27/16$ and exhibit a sequence of matrices that approaches it.

math.RA↗

arXiv2026

Variations on Majorization of Vectors and Connections to Determinantal Inequalities

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--P\'olya theorem for non-affine convex functions is closely linked to block structure in the associated doubly stochastic transformations. Motivated by this, we introduce $*$-majorization, a structured extension of majorization that respects prescribed block decompositions of vectors. This framework naturally corresponds to block diagonal doubly stochastic matrices and provides a refinement of the classical Hardy--Littlewood--P\'olya and Rado theorem. We show that such transformations are precisely the linear operators that preserve $*$-majorization, and we extend fundamental constructions such as $T$-transforms and convex combinations to this setting. In an application, we study the eigenvalue relations associated with the principal submatrices of positive definite matrices. Classical majorization does not, in general, capture determinantal inequalities such as those of Koteljanskii, whereas $*$-majorization provides a natural framework for structured comparisons of eigenvalue vectors. This leads to new insights into the interplay between majorization theory, determinantal inequalities, and spectral properties of matrices.

math.GM↗

arXiv2026

Determinant Bounds for $(n-1)$-Locally Positive Semidefinite Matrices

In this framework, the extremal case corresponds to the tightest nontrivial relaxation in this hierarchy, in which every proper principal submatrix is constrained to be positive semidefinite, while the global positive semidefiniteness condition is governed by the determinant. In this paper, we study the determinants of locally positive semidefinite matrices and derive sharp lower bounds on their determinants that quantify the gap between local and global positive semidefiniteness. We further obtain analogous extensions of classical determinant inequalities, including Fisher and Koteljanskii inequalities, providing tight lower bounds in each case. In a sense, these results quantify, via determinant bounds, how far the class of locally positive semidefinite matrices can be from being positive semidefinite.

math.OC↗

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↗

arXiv2019

Some geometric properties of the solutions of complex multi-affine polynomials of degree three

In this paper, we consider complex polynomials of degree three with distinct zeros and their polarization ((z1,z2,z3) with three complex variables. We show, through elementary means, that the variety P(z1,z2,z3)=0 is birationally equivalent to the variety z1z2z3 +1 = 0. Moreover, the rational map certifying the equivalence is a simple Möbius transformation. The second goal of this note is to present a geometrical curiosity relating the zeros of P(z,z,zk) for k = 1,2,3, where (z1,z2,z3) is arbitrary point on the variety P(z1,z2 z3) = 0.

math.CV↗

arXiv2012

Locally symmetric submanifolds lift to spectral manifolds

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold.

math.OC↗