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Bassam Mourad

Publications and source records attributed to Bassam Mourad.

11 recordsLinked to original sources
arXiv2026

Negative index, matchings, and nonnegative eigenvalues of tridiagonal stochastic matrices

We study negative eigenvalues of $n\times n$ stochastic matrices whose off-diagonal support is constrained by a sparse graph. The main tool is a matching-based inertia principle: if $G$ is bipartite with matching number $μ(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues counted with their multiplicities) is denoted by $ν_{-}(S) $, then \[ ν_{-}(S)\leq μ(G). \] In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ ν_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor. $ Consequently, after ordering the eigenvalues of $P$ in the decreasing order, we have $ λ_{\lceil n/2\rceil}(P)\geq0, \ \text{and hence} \ λ_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Next, we show that this tridiagonal bound is sharp in every dimension in both reducible and irreducible cases. Finally, we explore some possible extension and raise some open questions.

math.PR↗
arXiv2025

Jet Schemes, Newton Polygons and Continued Fractions

We study jet schemes of Newton non-degenerate plane curve singularities. We identify a subgraph of the graph of jet components and show that it can be constructed from walks on the lattice points in the first quadrant of the Cartesian plane. In particular, we determine all the irreducible components of the jet schemes. Furthermore, we prove that this subgraph encodes the embedded topological type of the curve singularity in the plane. Finally, we introduce a generating series defined in terms of the irreducible components of the jet schemes and their (co-)dimensions, and we prove that this series is rational and explicitly determine its poles.

math.AG↗
arXiv2023

On The Doubly Stochastic Realization Of Spectra

An $n$-list $λ:=\left(r; λ_2, \ldots, λ_n\right)$ of complex numbers with $r>0,$ is said to be realizable if $λ$ is the spectrum of $n\times n$ nonnegative matrix $A$ and in this case $A$ is said to be a nonnegative realization of $λ$. If, in addition, each row and column sum of $A$ equals $r$, then $λ$ is said to be doubly stochastically realizable and in such case $A$ is said to be a doubly stochastic realization for $λ$. In 1997, Guo proved that if $\left(λ_2,\ldots, λ_n\right)$ is any list of complex numbers which is closed under complex conjugation then there exists a least real number $λ_0$ with $\max\limits_{2\leq j\leq n}|λ_j|\leqλ_0\leq 2n\max\limits_{2\leq j\leq n}|λ_j|$ such that the list of complex numbers $\{ λ_1,λ_2,...,λ_n\}$ is realizable if and only if $λ_1\geq λ_0$. In 2020, Julio and Soto showed that the upper bound may be reduced to $(n-1)\max\limits_{2\leq j\leq n}|λ_j|$ in the case when at least one of the $λ_i$ is real. In this paper, we first describe an algorithm for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix $A$ to be cospectral to a doubly stochastic matrix $B$ and in this case $B$ is shown to be the unique closest doubly stochastic matrix to $A$ with respect to the Frobenius norm. Then, our next results slightly improve the upper bound for the nonnegative realization presented by Julio and Soto and in the case when none of the $λ_i$ is real, we also give an improvement of Guo's bound. Then, for doubly stochastic realizations, we obtain an upper bound that improves Guo's bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.

math.CO↗
arXiv2023

On a numerical construction of doubly stochastic matrices with prescribed eigenvalues

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method.

math.NA↗
arXiv2021

On an integral representation of the normalized trace of the $k$-th symmetric tensor power of matrices and some applications

Let $A$ be an $n\times n$ matrix and let $\vee^k A$ be its $k$-th symmetric tensor product. We express the normalized trace of $\vee^k A$ as an integral of the $k$-th powers of the numerical values of $A$ over the unit sphere $\mathbb{S}^{n}$ of $\mathbb{C}^{n}$ with respect to the normalized Euclidean surface measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over $\mathbb{C}^n$. As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in \cite{cuttler} concerning the monotonicity of products of complete symmetric polynomials. In the process, we give a solution to an open problem that was raised by I. Rovenţa and L. E. Temereanca in \cite{roventa}.

math.CO↗
arXiv2021

New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G. Soares in \cite{lemos}. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by R. Bhatia, Y. Lim and T. Yamazaki in \cite{Bhatia2}, and recently proved by T.H. Dinh, R. Dumitru and J.A. Franco in \cite{Dinh} for the Schatten p-norm with $1\leq p\leq \infty$.

math.FA↗
arXiv2020

On a new closed formula for the solution of second order linear difference equations and applications

In this note, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. As applications; we show that Binet's formula, in this case, is valid for negative integers as well. Finally, we find new summation formulas relating the elements of such sequences.

math.NT↗
arXiv2013

On the symmetric doubly stochastic matrices that are determined by their spectra

A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form $P^TAP$ for some permutation matrix P. The problem of characterizing such matrices is considered here. An almost the same but a more difficult problem was proposed by [ M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432 (2010) 2925-2927] as follows: Characterize all n-tuples $λ= (1,λ_2,...,λ_n)$ such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum $λ.$ In this short note, some general results concerning our two problems are first obtained. Then, we completely solve these two problems for the case n = 3. Some connections with spectral graph theory are then studied. Finally, concerning the general case, two open questions are posed and a conjecture is introduced.

math.CO↗