In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson involving Hessenberg matrices is consequently generalized.

Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda =\rho \left(A\right),{\lambda }_2,...,{\lambda }_n$. Fiedler and others have shown that $det(\lambda I-A)\le {\lambda }^n-{\rho }^n$, for all $\lambda >\rho$, with equality for any such $\lambda$ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i=1,...,n-1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det(\lambda I-A)+{\sum }_{i=1}^{n-1}{\rho }^{n-2i}\left|a_i\right|{(\lambda -\rho )}^i\le {\lambda }^n-{\rho }^n$, for all $\lambda \ge \rho$. We use this inequality to derive the inequality that: ${\prod }_2^n(\rho -{\lambda }_i)\le {\rho }^{n-2}{\sum }_{i=2}^n(\rho -{\lambda }_i)$. In the spirit of a celebrated conjecture due to Boyle-Handelman,...

Given two measured spaces $(X,\mu )$ and $(Y,\nu )$, and a third space $Z$, given two functions $u(x,z)$ and $v(x,z)$, we study the problem of finding two maps $s:X\to Z$ and $t:Y\to Z$ such that the images $s\left(\mu \right)$ and $t\left(\nu \right)$ coincide, and the integral ${\int }_{X}u(x,s\left(x\right))d\mu -{\int }_{Y}v(y,t\left(y\right))d\nu$ is maximal. We give condition on $u$ and $v$ for which there is a unique solution.

Given two measured spaces $(X,\mu )$ and $(Y,\nu )$, and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps $s:X\to Z$ and $t:Y\to Z$ such that the images $s\left(\mu \right)$ and $t\left(\nu \right)$ coincide, and the integral ${\int }_{X}u(x,s\left(x\right))d\mu -{\int }_{Y}v(y,t\left(y\right))d\nu$ is maximal. We give condition on u and v for which there is a unique solution.

Mediante integrali multipli agevoli per il calcolo numerico vengono espressi il valore assoluto di un determinante qualsiasi e le formule di Cramer.

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise...

The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known...

Companion matrices of the second type are characterized by properties that involve bilinear maps.