International Conference on Linear Algebra and its Applications
(ICLAA 2023, Manipal, India)

Let $ M, D, K$ be $n\times n$ matrices such that $M$ and $K$ are positive semidefinite and $D$ is Hermitian. Consider the quadratic eigenvalue problem (QEP) $$P(\lambda)v := (\lambda^2 M + \lambda D + K)v= 0$$associated with a damped mass-spring system. The mass matrix $M$ is usually positive definite. However, there are applications where $M$ is positive semidefite and $K$ is positive definite. In such a case, $\infty$ is an eigenvalue of $P(\lambda)$. The presence of an infinite eigenvalue poses computational difficulty in solving the QEP. We describe a structure-preserving deflation strategy that deflates the eigenvalue $\infty$ and produces a reduced order model $ \widehat{P}(\lambda) := \lambda^2\widehat{M} + \lambda \widehat{D} + \widehat{K},$ where both $\widehat{M}$ and $\widehat{K}$ are positive definite and $\widehat{D}$ is Hermitian. We also discuss hyperbolicity and inertia of $P(\lambda).$ The damping matrix $D$ usually has low rank because of which $P(\lambda)$ has undamped (purely imaginary) eigenvalues. We analyze the effect of damping on the purely imaginary eigenvalues of $P(\lambda)$ and present a structure-preserving deflation strategy that deflates all the purely imaginary eigenvalues of $P(\lambda)$ and produces a reduced order model $ \widetilde{P}(\lambda) := \lambda^2\widetilde{M} + \lambda \widetilde{D} + \widetilde{K}.$ We also analyze removal of purely imaginary eigenvalues of $P(\lambda)$ via low rank perturbation of the damping matrix $D$.

A two-stage sequential procedure to estimate the parameters of a cumulative exposure model under an accelerated testing scenario is discussed. We focus on a step-stress model where the stress level is updated after a pre-specified number of failures occur, which is also random. This is termed as the ‘random stress change time’ in the literature. To obtain maximum precision, a certain variance optimality criterion is applied. A pseudo real data example from reliability studies is also analyzed to outline the performance of the proposed methodology.

The talk is a survey of some applications of matrix theory to graph theory and of graph theory to matrix theory.

Here are some examples: Every doubly nonnegative realization of a graph G is completely positive if and only if the line graph of G is perfect.

Graphs of pyramids are determined by their spectrum.

Construction of pairs of non regular non isomorphic graphs that are cospectral with respect to the adjacency matrix, the Laplacian, the sign less Laplacian and the normalized Laplacian.

Given a positive semi-definite matrix $G$ and another matrix $X$, we present a method of obtaining disjoint symmetric matrices $G_X$ and $G − G_X$ where range of $G_X$ is the intersection of the ranges of $G$ and $X$. $G_X$ is called a section of $G$. Properties of $G_X$ and its applications in quadratic minimization and Gauss Markov Model are given in this talk.

A matrix is called totally positive (TP) if all of its minors are positive. For over a century this class of matrices (and extensions to kernels, sequences, and distributions) has been studied and interesting applications continue to be developed. Recent interest on this topic stems from classifying entry-wise transformations that preserve TP matrices. While studying a particular instance of this problem we were lead to a new determinantal inequality associated with TP matrices. In verifying this inequality, we developed a technique which has proven useful for investigating families of additive and multiplicative determinantal inequalities (and identities) for TP matrices. I will discuss some of the history and recent research, which is joint with a PIMS PDF Dr. P.K. Vishwakarma.

The initial results on necessary and sufficient conditions for BLUEs of estimable functions of parameters in a linear fixed effect model to remain un-altered despite changes in error covariance structure are given by Rao (1971). Rao’s results can be extended to any full rank or singular error covariance so that not only are the BLUEs in the linear model retained, but so is their covariance or their errors sum of squares. For submodels, where a full linear model is made smaller by reducing the number of regressors, block diagonal or diagonal matrices can provide insight into conditions for the full model and its entire set of submodels each to retain some or all of their BLUEs (Haslett, Istalo, Markiewicz, et al. (2023); Haslett, Puntanen (2023)) as well as retain the covariance of these BLUEs. Such error covariance changes can produce a wide variety of new cloned data sets, none of which seem to have any resemblance to the original data but which nevertheless have the same BLUEs and BLUE covariances for each submodel as for the original data. The results have application in data confidentiality, encryption, genetics and residual analysis.

This talk will be related to the study of matrix algebras having the property that each singular element can be expressed as a product of idempotents – a subject that had and has been of interest to several mathematicians. For example, Howie was the first who showed that a non-invertible map on a finite set can be expressed a product of idempotent maps. This led to consider specific algebras having this property. Erdos, Victoria Gould, Fountain, Laffey, Lam, Leroy, O’Meara and many others studied as to when a given associative algebra has this property. Besides its intrinsic interest of studying structure of such algebras , this is also related to a long standing open question recently noted by Leroy-Jain, whether a von Neumann regular ring is always separative.

In this talk, the class of bicyclic graphs with a unique perfect matching have been taken into consideration. This class is designated as $\mathcal{B}$. We identify all the graphs $B$ in $\mathcal{B}$ such that the diagonal entries of the inverse of the adjacency matrix of $B$ are all zero. We provide criteria for the adjacency of two vertices in the inverse of the graph $B\in\mathcal{B}$. Additionally, we describe the structure of those graphs in $\mathcal{B}$, whose inverse is a tree or a unicyclic graph.

The $\textit{eccentricity matrix}$ $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-$\textit{eigenvalues}$ of $G$. It is well known that the distance matrices of tress are invertible and the determinant depends only on the number of vertices. We show that the eccentricity matrix of tree $T$ is invertible if and only if either $T$ is star or $P_4$. Also we show that any tree with odd diameter has $4$ distinct $\mathcal{E}$-eigenvalues, and any tree with even diameter has the same number of positive and negative $\mathcal{E}$-eigenvalues (which is equal to the number of ’diametrically distinguished’ vertices). Finally, we will discuss about the trees with $\mathcal{E}$-eigenvalues are symmetric with respect to the origin.

In this talk we discuss finite and infinite totally positive (TP) matrices, including the origins of their study — which date back morally to the 1600s, and mathematically to the 1800s. After discussing examples and basic properties of these matrices, we will see early results by Laguerre, Fekete (in correspondence with Polya), Schoenberg, and Motzkin. These results characterize total positive — and more broadly, strictly sign regular (SSR) — $n \times n$ matrices, via the variation diminishing property (VD) for all vectors of length $n$. We will end with recent results of P.N.\ Choudhury (including with coauthors) which reveal other characterizations of TP. In particular, to deduce that a matrix is TP/SSR, it suffices to consider the VD property for just a single vector for each square submatrix.

A square nonnegative matrix T is called stochastic if all of its row sums are equal to 1. Under mild conditions, it turns out that there is a positive row vector w^T (called the stationary distribution for T) whose entries sum to 1 such that the powers of T converge to the outer product of w^T with the all-ones vector. Further, the nature of that convergence is governed by the eigenvalues of T.

In this talk we explore how the stationary distribution for a stochastic matrix exerts an influence on the corresponding eigenvalues. We do so by considering the region in the complex plane comprised of all eigenvalues of all stochastic matrices with a given stationary distribution. We establish a few properties of that region, and of the variant that arises by considering the so-called reversible stochastic matrices. For the reversible version of the problem, the graphs associated with the reversible stochastic matrices are a useful tool.

In 1876 H. J. S. Smith defined an LCM matrix as follows: let $S=\{x_1,x_2,\ldots,x_n\}$ be a set of positive integers with $x_1$<$x_2$<$\cdots$<$x_n$. The LCM matrix $[S]$ on the set $S$ is the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set $S$ (which means that $\gcd(x_i,x_j)\in S$ for all $i,j\in\{1,2,\ldots,n\}$), suffices to guarantee the invertibility of the matrix $[S]$. However, a few years later this conjecture was proven false first by Haukkanen et al. and then by Hong. It turned out that the conjecture holds only on GCD closed sets with at most 7 elements but not in general for larger sets. The GCD closedness of the set $S$ also makes it possible to study the invertibility and inertia of the corresponding LCM matrix by using a certain lattice theoretic method, see Altinisik, E. et al.(2017), Haukkanen, P. et al. (2020), M. Mattila et al. (2023) and Korkee, I. et al. (2018). However, this same lattice theoretic method can be applied to study the so called power LCM matrices on a GCD closed set $S$, which are simply the Hadamard powers of the ordinary LCM matrix $[S]$. In this presentation we shall make use of this lattice theoretic method and present some interesting results about singular power LCM matrices. The content of this talk is based on the articles Haukkanen, P. et al. (2015) and M. Mattila et al. (2023).

Let $\lambda_1(T)\ge \lambda_2(T)\ge \cdots \ge\lambda_n(T)$ be the eigenvalues of an $n$-vertex tree $T$. Trees for which $\lambda_1(T)$ or $\lambda_2(T)$ is largest or smallest possible among all $n$-vertex trees have been classified. In this talk the speaker will discuss extremal trees for linear combinations
$\alpha \lambda_1(T)+ \beta \lambda_2 (T)$, where $\alpha, \beta \in \mathbb{R}$. This is joint work with Hitesh Kumar, Shivaramakrishna Pragada, and Harmony Zhan.

In this talk a new matrix class $\bar{L}(d)$ is discussed which are obtained as a limit of a sequence of $L(d)$ matrices introduced by Garcia. Various properties and characterizations of this new matrix class are discussed. Role of this new matrix class in pivoting algorithms are also explained. An application of this new matrix class that arises from general quadratic programs and polymatrix games is also discussed. Finally, an example is presented related to the existence of equilibrium in polymatrix games.

In this talk I will consider my Indian experiences since 1980s: meetings with statisticians, mathematicians and the scenary, with several photographic glimpses.

A closed convex cone $K$ with the origin as an extreme point in $\mathbb{R}^n$ induces a partial order in $\mathbb{R}^n$. A cone $K$ is called minihedral, if the partial order induces the lattice structure. Any lattice cone with interior in $\mathbb{R}^n$ admits a basis with the property that the Cone is simply the collection of all nonnegative linear combinations of basis vectors. It is through this theorem of Yudin, that we show that any linear transformation $A$ which leaves this cone invariant with a fixed vector in the interior of the cone can be represented as a stochastic matrix with respect to a suitable basis in $K$. The theorem has extensions to stochastic operators and coordinate free representation of Positive operators as doubly stochastic matrices, with appropriate conditions.

This is a joint work with S. Natarajan and K. Viswanath

Will be updated soon.

Let X be an infinite dimensional Banach space. In this talk, we give an exposition on the problem of how finite dimensional perturbation of a subspace effects several proximinality related properties of the subspace. We also illustrate conditions on a Banach space X so that all finite codimensional subspaces of a certain type of proximinality occur as finite dimensional perturbations of some standard type of subspaces .

Penalized estimation methods will be considered (Ridge estimation). If there is a model including parameters and an estimation function, e.g., the least squares function, so called penalized estimators can be obtained. This means that the estimation function is modified by adding some ”fitting” term. However, in this presentation of the subject, restrictions are put on the parameters instead of the estimation function which in turn also will lead to a penalized estimation function. Indeed the two different approaches are very similar. The results will often be the same but interpretations can differ. In some way, from a likelihood point of view, it is more natural to put restrictions on the parameters in a model than on the estimation function. Our approach is based on convex optimization theory.

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 $\eta(G)$ and indicate open problems and directions for further research.

Local order isomorphisms of matrix and operator domains will be discussed. A connection with Loewner’s theorem and the fundamental theorem of chronogeometry will be explained. The first one characterizes operator monotone functions while the second one describes the general form of bijective preservers of light-likeness on the classical Minkowski space.

Let $m,n \ge 2$ be natural numbers. If $\xi$ and $\eta$ are vectors of length $m$ and $n$ respectively, their tensor product $\zeta = \xi \otimes \eta$ can be identified with the $ m \times n$ matrix $A = \xi \eta^t$, where $t$ is the transpose, leading to treating the tensor product $\mathcal{H}=\mathbb{C}^m \otimes \mathbb{C}^n$ as the linear space $\mathcal{M}_{m,n}$ of complex $m \times n$ matrices. Vectors $\zeta$ in $\mathcal{H}$ of the form $\xi \otimes \eta$ are called product or separable vectors and all others are called entangled vectors, the latter lot corresponds to matrices with rank $\ge 2$. Quantum Entanglement is an important and useful concept in Quantum Information Theory (QIT). Multipartite set-up $\mathcal{H} = \overunderset{k}{j=1}{\otimes} \mathcal{H}_j$ with $\mathcal{H}_j=\mathbb{C}^{d_j}$, $d_j \ge 2$ for $1 \le j \le k$ can be conveniently described as the linear space of complex polynomials in $k$ variables $\mathbf{X}=(X_j)^k_{j=1}$ with degree in $X_j$ less than or equal to $d_j-1$ for $1\le j \le k$ and it provides a simple approach to study quantum entanglement. Subspaces $\mathcal{S}$ of $\mathcal{H}$ that are completely entangled in the sense that all non-zero vectors in $\mathcal{S}$ are entangled have been well studied and two basic methods for their construction come from use of Unextendable Product Bases (UPB) and use of Van der Monde matrices by taking orthogonal complements of their linear spans $\mathcal{U}$ and $\mathcal{F}$ respectively. Examples of $\mathcal{U}$ having only finitely many well-specified product vectors and linear span of a single vector from UPB or $\mathcal{F}$ and the corresponding $\mathcal{S}$ with the same property are of interest in QTT. The talk will illustrate the related development over the decades including the recent one by us and others.

For real square matrices, it is known that, if $A$ is a singular irreducible $M$-matrix, then the only nonnegative vector that belongs to the range space of is the zero vector. In this talk, we present an analogue of this result for the Lyapunov and the Stein operators, on the space of real symmetric matrices.

Associated to the $\textit{Four point condition}$ (4PC henceforth) and a tree $T$ on $n$ vertices are two matrices, the $ \text{Max}_{T} $ and the $ \text{Min}_{T} $. Both are $\binom{n}{2} \times \binom{n}{2}$ matrices.

The rows and columns of both matrices are indexed by pairs $\{i,j\}$ of vertices. The entry of the matrix $ \text{Min}_{T} $ in the row indexed by $\{i,j\}$ and column $\{k,l\}$ is the minimum value among the three terms in $S_{i,j,k,l} = \{d_{i,j} + d_{k,l}, d_{i,l} + d_{j,k}, d_{i,k} + d_{j,l} \}$. Replacing the word “minimum” by “maximum” in the previous sentence gives us the $ \text{Max}_{T} $ matrix.

Though some work has been done about the $ \text{Min}_{T} $ matrix, there is not much known about the $ \text{Max}_{T} $ matrix. We show results about the rank, give an algorithm that outputs a basis $B$ for the row space of $ \text{Max}_{T} $ and give the determinant of $ \text{Max}_{T}[B,B] $.

This is joint work with Ali Azimi, Rakesh Jana and Mukesh Nagar.

Inspired by Biane’s map from involutions to Motzkin paths we define a map from subspaces to Motzkin paths, using Gauss elimination. As an application we revisit Vogt and Voigt’s remarkable solution to the Greene-Kleitman problem of constructing an explicit symmetric chain decomposition of the subspace lattice. We give a highly elegant and simpler reformulation of Vogt and Voigt’s solution.

This is joint work with Jonathan Farley.