International Conference on Linear Algebra and its Applications (ICLAA 2021)

Let $ U \subset \mathbb{C}$ be open and $T: U \longrightarrow \mathbb{C}^{n\times n}$ be holomorphic. Consider the nonlinear eigenvalue problem (NEP): $$ \text{ Find } \lambda \in U \text{ and a nonzero } v \in \mathbb{C}^n \text{ such that } T(\lambda) v = 0.$$ NEPs arise in many applications in science and engineering. Computing solution of an NEP is a challenging task and various methods for solving NEPs have been proposed in recent years. We present a trace-moment based method for computing eigenvalues of $T(\lambda).$ If a region in $U$ contains $\ell$ distinct eigenvalue $\lambda_1, \ldots, \lambda_\ell$ of $T(\lambda)$ then by utilizing trace-moments of $T(\lambda)$ we construct an $\ell\times \ell$ Hankel pencil $L(\lambda) := \widehat{H}- \lambda H$ such that $\lambda_1, \ldots, \lambda_\ell$ are simple eigenvalues of $ L(\lambda).$ Hence by solving the linear eigenvalue problem $L(\lambda) v =0$ we obtain the eigenvalues $\lambda_1, \ldots, \lambda_\ell$ of $T(\lambda).$

Let $G$ be a simple graph of order $n$. The nullity $\eta(G)$ of $G$ is the algebraic multiplicity of $0$ as an eigen value of the adjacency matrix of $G$. A graph $G$ with $\eta(G) > 0$ is called a singular graph. For a bipartite graph $G$ (corresponding to an alterant hydrocarbon) if $\eta(G) > 0$, then it indicates that the molecule which such a graph represents is unstable. A characterization of graphs for which $\eta(G) > 0$ still remains open. In this talk we present a survey of results on $η(G)$ and indicate open problems and directions for further research.

Let $S$ be a set of nonnegative numbers. A matrix $A$ is completely positive over $S$ if it can be decomposed as $A=BB^T$, where the entries of $B$ are in $S$. The talk is a survey of results on completely positivity over $S$ in the cases $S=R,\;S=Q\; S=Z$ and $S=\{0,1\}$.

Given a matrix $M$ and a vector $q$, the Linear Complementarity Problem(LCP($M,q$) is to find a solution $z$ to $$ [Mz + q \ge 0\, z^{T}(Mz+q) = , z \ge 0] $$ Lemke’s Algorithm applied to LCP($M,q+\theta p$) fetches solutions to the LCP for all $\theta \in [\theta_{0}, \infty)$ for some $\theta_{0}$ and $p > 0$. We express $\theta$ as a function of time index $t$. Solution $z$ obtained from Lemke’s Algorithm is a continuous function of $t$. A variant of Lemke’s Algorithm is proposed using the continuity property. Some results on characterization of $Q$-matrices are obtained.

The inverse eigenvalue problem for graphs has become a central research enterprise for the past 30 years with many exciting and interesting advances and related applications. One aspect of this problem is studying the fewest possible eigenvalues associated with a given graph.

This presentation will provide a broad survey of current results involving this curious and important graph parameter, which is typically labeled as $q(G)$ for a given graph $G$. In particular, I will discuss the cases when $q(G) \geq |G|-1$ and when $q(G)=2$, along with other intriguing results on $q$ for various families of graphs. I will also touch upon recent variants of the parameter $q$ connected with certain `strong’ properties of matrices.

By using semigroup techniques, J. Araújo and F.C. Silva proved that a matrix B with coefficients in a division ring D is a product of conjugates of any matrix A with rank(B) smaller or equal to rank(A). We prove this result over an algebraically closed field, in an elementary way suitable even for undergraduate students.

A block is said to be pendant if it has exactly one point of articulation. Suppose that we are given a collection blocks and we construct all possible connected graphs with these blocks keeping the number of pendant blocks fixed. In this talk, we describe the structure of the graphs that minimize the algebraic connectivity among all such graphs. As an application, we conclude that over all such graphs made with the given blocks, the algebraic connectivity is minimum for a graph whose block structure is a path.

Given a finite simple connected graph $G = (V,E)$, we introduce a novel invariant which we call its blowup-polynomial $p_G((n_v)_{v \in V})$. To do so, we compute the determinant of the distance matrix of the graph blowup, obtained by taking $n_v$ copies of the vertex $v$, and remove an exponential factor. First: we show that as a function of the sizes $n_v$, $p_G$ is a polynomial, is multi-affine, and is real-stable. Second: we show that the multivariate polynomial $p_G$ is intimately related to the characteristic polynomial $q_G$ of the distance matrix $D_G$, and that it fully recovers $G$ whereas $q_G$ does not. Third: we obtain a novel characterization of the complete multi-partite graphs, as precisely those whose “homogenized” blowup-polynomials are Lorentzian/strongly Rayleigh. (Joint with Projesh Nath Choudhury.)

We consider the spread of an infectious disease, modelled as a network of patches (to reflect a heterogeneous environment) with movement between patches. The invasibility of the disease can be measured by the network reproduction number $R_0$, which turns out to be the spectral radius of a certain nonnegative matrix. In this talk we work with an approximation to $R_0$; that approximation applies in the case that the time scale of movement is substantially larger than the time scale of the disease dynamics. We investigate how perturbations in the network structure affect the value of $R_0$, and discuss the changes to the network that yield the largest decrease in $R_0$. Throughout we use techniques from matrix analysis, and adopt perspectives from combinatorics.

We consider the general linear model $y = X \beta + \varepsilon$ supplemented with the new unobservable random vector $ y_{*}$,coming from $ y_{*} = X_{*}\beta + \varepsilon_{*}$, where the covariance matrix of $y_{*}$ is known as well as the cross-covariance matrix between $y_{*}$ and $y$. A linear statistic $F y$ is called linearly sufficient for $X_{*} \beta$ if there exists a matrix $A$ such that $A F y$ is the best linear unbiased estimator, BLUE, for $X_{*} \beta$. The concept of linear sufficiency with respect to a predictable random vector is defined in the corresponding way but considering the best linear unbiased predictor, BLUP, instead of BLUE. In this paper, we consider the linear sufficiency of $F y$ with respect to $ y_{*}$, $X_{*} \beta$, and $\varepsilon_{*}$.

Let $F=[f_{ij}]$ be an $n \times n$ symmetric matrix. Define $d_{ij}:=f_{ii}+f_{jj}-2f_{ij}$. Now, the matrix $D=[d_{ij}]$ is a called a Euclidean distance matrix (EDM). EDMs have several interesting properties. We introduce a simple generalization of a EDM. Fix $a,b>0$. Define $$E=[a^{2}g_{ii} + b^{2}g_{jj}-2ab g_{ij}],$$ where $G=\left[ g_{ij} \right]$ is a positive semidefinite matrix such that $g\mathbf{1}=0$, where $\mathbf{1}$ is the vector of all ones in $\mathbb{R}^{n}$. We call $E$ a generalized EDM. Despite $E$ being a non-symmetric matrix, many of the interesting properties of a EDM can be extended to a generalized EDM. All these properties will be discussed the talk.

An important direction of investigation in Operator theory of Banach spaces, is, to perform standard Banach space theoretic operations on spaces of operators and ask if the resulting object is again a space of operators (possibly between different Banach spaces). In this talk, we tackle this problem for the well-known geometric operation, Birkhoff-James orthogonality.
For non-reflexive Banach spaces, $X,Y$, for a closed subspace $M$ of operators, we investigate Birkhoff-James orthogonality of an operator $T$ to $M$ with that of $T^{**}$ with an appropriate subspace of $M^{**}$.

Ryser design can be described as a binary square matrix $A$ for which $A^tA$ equals the sum of a diagonal matrix and a multiple of $J$, the all $1$ matrix with the additional requirement that $A$ has at least two different column sums (equivalently two different row sums). This talk will discuss two different proofs of the Ryser-Woodall theorem (on Ryser designs), the first of which is by Ryser that is combinatorial in nature and the second by Ionin and M.S. Shrikhande that is linear algebraic. This talk will give an account of the known results on Ryser’s conjecture and some recent work in this direction.

In both univariate and multivariate analysis, residuals are used for model diagnostics. In this presentation we consider the MANOVA and the GMANOVA-MANOVA models and different matrix residuals are established. The interpretation of the residuals is discussed and several properties are verified. A realistic, similar to some real studies, but artificial data will be analysed to illustrate how the residuals can be used in model validations.

Let $\mathcal{S}^n$ denote the space of real symmetric matrices of order $n \times n.$ A linear operator $T$ on $\mathcal{S}^n$ is said to be a $P$-operator, if the following implication holds for every $X \in \mathcal{S}^n$: $$XT(X)=T(X)X \preceq 0 \Longrightarrow X=0,$$ where, for $Y \in \mathcal{S}^n$ we use $Y \succeq 0$ to denote that $Y$ is positive semidefinite, i.e., $u^tYu \geq 0$ for all $u \in \mathbb{R}^n$. Also, $-Y \succeq 0$ is denoted by $Y \preceq 0$.

Let $A \in \mathbb{R}^{n \times n}$ be fixed. The following three maps have been well studied, in the context of the semidefinite linear complementarity problems (SDLCP).

The Lyapunov transformation $L_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, is defined by $$L_A(X)=AX+XA^T,$$ the Stein transformation $S_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, by $$S_A(X)= X-AXA^T$$ and the multiplicative transformation $M_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, by $$M_A(X)= AXA^T.$$ In [M. S. Gowda, Y. Song(2000), Theorem 5], it is proved that $L_A$ has the $P$-property iff $A$ is positive stable (meaning, all the eigenvalues of $A$ have positive real part). Also, in [M. S. Gowda, T. Parthasarathy(2000), Theorem 6], it is shown that $S_{A}$ has $P$-property iff $A$ is Schur stable (meaning, all the eigenvalues of $A$ lie in the open unit disk). Finally, it is shown in [P. Bhimashankaran, T. Parthasarathy, A. L. N. Murthy, G. S. R. Murthy(2012), Theorem 17] that $M_A$ has $P$-property iff $A$ is either positive definite or negative definite. It is important to observe that these results have some interesting and nontrivial implications to SDLCP.

A linear operator $T$ on $\mathcal{S}^n$ is a $P_{\#}$-operator [introduced in M. Rajesh Kanna, K. C. Sivakumar(2014), Definition 1.3], if for every $X \in \mathcal{S}^n$ we have: $$X \in R(T), XT(X)=T(X)X \preceq 0 \Longrightarrow X=0.$$ For $x,y \in \mathcal{S}^n,$ define $X\circ Y:=\frac{1}{2}(XY+YX).$ We say that the operator $T$ has the Jordan-$P$-property if $$X\circ L(X) \preceq 0 \Longrightarrow X=0.$$
$T$ is said to possess the Jordan-w-$P$-property if $$X\circ L(X) \preceq 0 \Longrightarrow L(X)=0.$$
Finally, $T$ is said to satisfy the w-$P$-property if $$XL(X) =L(X)X, ~X\circ L(X) \preceq 0 \Longrightarrow L(X)=0.$$ The three notions given in the previous para were proposed in [J Tao(2009)]. The results stated earlier for the Lyapunov, Stein and the multiplicative transformations, have been shown to have analogues for operators satisfying any of the four latter properties defined above (see, for instance [I. Jeyaraman, Kavita Bisht, K. C. Sivakumar(2017)] and [J. Tao(2009)]).

In this talk, first we give a brief survey. We then present some new relationships between operators belonging to these classes.

Let $T$ be a tree on $n$ vertices and let $\mathscr{L}^T_q$ be the $q-$analogue of its Laplacian. For a partition $\lambda \vdash n$, let the normalized immanant of $\mathscr{L}^T_q$ indexed by $\lambda$ be denoted as $\overline{\text{Imm}}_\lambda(\mathscr{L}^T_q)$. A string of inequalities among $\overline{\text{Imm}}_\lambda(\mathscr{L}^T_q)$ is known when $\lambda$ varies over hook partitions of $n$ as the size of the first part of $\lambda$ decreases. In this work, we show a similar sequence of inequalities when $\lambda$ varies over two row partitions of $n$ as the size of the first part of $\lambda$ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of $\mathfrak{S}$ indexed by two row partitions.
Our proof can be interpreted using the combinatorics of Riordan paths and our main lemma admits a nice probabilisitic interpretation involving peaks at odd heights in generalized Dyck paths or equivalently involving special descents in Standard Young Tableaux with two rows. As a corollary, we also get inequalities between $\overline{\text{Imm}}_\lambda(\mathscr{L}^{T_1}_q)$ and $\overline{\text{Imm}}_\lambda(\mathscr{L}^{T_1}_q)$ when $T_1$ and $T_2$ are comparable trees in the $\text{GTS}_n$ poset and when $\lambda_1$ and $\lambda_2$ are both two rowed partitions of $n$, with $\lambda_1$ having a larger first part than $\lambda_2$.

The adjacency matrix of the $n$-cube and the closely related tridiagonal matrix of Mark Kac have an elegant spectral theory that arise in a variety of applications. This paper defines $q$-analogs of these two matrices and studies their spectral theory.

We consider two applications: a $q$-analog of the random walk on the $n$-cube and a $q$-analog of the product formula for the number of rooted spanning trees of the $n$-cube.