Integer completely positive matrices of order two
We show that every integer doubly nonnegative $2 \times 2$ matrix has an integer cp-factorization.
math.OC↗
arXiv subjects
Helena Šmigoc
Publications and source records attributed to Helena Šmigoc.
Nonnegative Factorization of a Data Matrix as a Motivational Example for Basic Linear Algebra
We present a motivating example for matrix multiplication based on factoring a data matrix. Traditionally, matrix multiplication is motivated by applications in physics: composing rigid transformations, scaling, sheering, etc. We present an engaging modern example which naturally motivates a variety of matrix manipulations, and a variety of different ways of viewing matrix multiplication. We exhibit a low-rank non-negative decomposition (NMF) of a "data matrix" whose entries are word frequencies across a corpus of documents. We then explore the meaning of the entries in the decomposition, find natural interpretations of intermediate quantities that arise in several different ways of writing the matrix product, and show the utility of various matrix operations. This example gives the students a glimpse of the power of an advanced linear algebraic technique used in modern data science.
math.HO↗
Diagonal elements in the Nonnegative Inverse Eigenvalue Problem
We say that a list of complex numbers is "realisable" if it is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterising all realisable lists. Although the NIEP remains unsolved, it has been solved in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. For a given spectrum of this type, we show that a list of nonnegative numbers may arise as the diagonal elements of the realising matrix if and only if these numbers satisfy a remarkably simple inequality. Furthermore, we show that realisation can be achieved by the sum of a companion matrix and a diagonal matrix.
math.SP↗
An extension of the Hermite-Biehler theorem with application to polynomials with one positive root
If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots interlace. This bound depends on the number of roots of $f$ which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive roots of a polynomial in terms of the number of sign changes in its coefficients. We use our extension of the Hermite-Biehler Theorem to give an inverse rule of signs for polynomials with one positive root.
math.CA↗
The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph
In this paper we introduce a parameter $Mm(G)$, defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph $G$. We compute $Mm(G)$ for several families of graphs.
math.SP↗
Families of Newton-like inequalities for sets of self-conjugate complex numbers
We derive families of Newton-like inequalities involving the elementary symmetric functions of sets of self-conjugate complex numbers in the right half-plane. These are the first known inequalities of this type which are independent of the proximity of the complex numbers to the real axis.
math.CA↗
Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem
We say that a list of real numbers is "symmetrically realisable" if it is the spectrum of some (entrywise) nonnegative symmetric matrix. The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all symmetrically realisable lists. In this paper, we present a recursive method for constructing symmetrically realisable lists. The properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years, starting with the work of Fiedler in 1974. We show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing.
math.SP↗
The effect of assuming the identity as a generator on the length of the matrix algebra
Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest number $k$ such that words of length not greater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets $\mathcal S$ and counted as a word of length $0$. In this paper we discuss how the problem changes if this assumption is removed.
math.RA↗
Graphs that allow all the eigenvalue multiplicities to be even
Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue problem for a graph $G$ is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in $S(G).$ This question is, in general, hard to answer and several variations were studied, most notably the minimum rank problem. In this paper we introduce the problem of determining for which graphs $G$ there exists a matrix in $S(G)$ whose characteristic polynomial is a square, i.e. the multiplicities of all its eigenvalues are even. We solve this question for several families of graphs.
math.SP↗
Power series with positive coefficients arising from the characteristic polynomials of positive matrices
Let A be an nxn (entrywise) positive matrix and let f(t)=det(I-t A). We prove that there always exists a positive integer N such that 1-f(t)^{1/N} has positive coefficients.
math.SP↗
Constructing New Realisable Lists from Old in the NIEP
Given a list of complex numbers σ:=(λ_1,λ_2,...,λ_m), we say that σ is realisable if σ is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is the problem of categorising all realisable lists. Given a realisable list (ρ,λ_2,λ_3,...,λ_m), where ρ is the Perron eigenvalue and λ_2 is real, we find families of lists (μ_1,μ_2,...,μ_n), for which (μ_1,μ_2,...,μ_n,λ_3,λ_4,...,λ_m) is realisable. In addition, given a realisable list (ρ,α+iβ,α-iβ,λ_4,λ_5,...,λ_m), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ_1,μ_2,μ_3,μ_4), for which (μ_1,μ_2,μ_3,μ_4,λ_4,λ_5,...,λ_m) is realisable.
math.SP↗
Solution Theory for Systems of Bilinear Equations
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from elementary operations. This idea is used to identify bilinear systems that are solvable for all right hand sides and to understand solvability when the number of equations is large or small.
math.RA↗