Schedule for: 17w2668 - Special Western Canada Linear Algebra meeting

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

We survey some results concerning the use of numerical ranges to study dilation theory of operators. Connections to problems arising in quantum information science will be mentioned.

Let $A$ be a matrix with spectrum $\sigma(A)$. The Kreiss Matrix Theorem (KMT), a well-known fact in applied matrix analysis, gives estimates of upper bounds for $\|A^n\|$ if $\sigma(A)$ is in the unit disc, or for $\|\text{e}^{tA}\|$ if $\sigma(A)$ is in the left-half plane based on the resolvent norm, or equivalently the pseudospectra. In this talk, we will discuss the extension of this celebrated theorem to an arbitrary holomorphic function. We will first briefly talk about the norm behavior of matrices that have identical or super-identical pseudospectra. Next we will focus on some generalizations of KMT for holomorphic functions on a general complex domain.

A square matrix can be reduced to simpler form via similarity transformations. Here ``simpler form'' may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Similar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, unimodular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an applications, we will illustrate how to use these reduced forms to solve parameterized linear systems.

Finite, discrete, time-homogeneous Markov chains are frequently used as a simple mathematical model of real-world dynamical systems. In many such applications, an analysis of clustering behaviour is desirable, and it is well-known that the eigendecomposition of the transition matrix $T$ of the chain can provide such insight. In a recent paper (see [1]), a method is presented for determining clusters from a subdominant real eigenvalue $\lambda$ of $T$ which is close to the spectral radius 1. In this talk, we extend the method to include an analysis for complex eigenvalues of $T$ which are close to 1. Since a real spectrum is not guaranteed in most applications, this is a valuable result in the area of spectral clustering in Markov chains. This is joint work with Emanuele Crisostomi, Mahsa Faizrahnemoon, Steve Kirkland, and Robert Shorten. [1] Emanuele Crisostomi, Stephen Kirkland, and Robert Shorten. A Google-like model of road network dynamics and its application to regulation and control. $\textit{International Journal of Control}$, 84(3):633--651, 2011.

The $\textit{principal minor assignment problem}$ asks the following question: Can we find an $n \times n$ matrix having prescribed principal minors? As a simplification of this problem, researchers associated a sequence with a symmetric (or complex Hermitian) matrix, which they defined as follows: The $\textit{enhanced principal rank characteristic sequence} (\textit{epr-sequence})$ of an $n \times n$ symmetric (or complex Hermitian) matrix $B$ is $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is $\tt{A}$ (respectively, $\tt{N}$) if all (respectively, none) the principal minors of order $k$ are nonzero; if some (but not all) are nonzero, then $\ell_k = \tt{S}$. As a refinement of the epr-sequence, the present author recently introduced the $\textit{signed enhanced principal rank characteristic sequence} (\textit{sepr-sequence})$ of an $n \times n$ Hermitian matrix, denoted $t_1t_2 \cdots t_n$, where $t_k$ is either $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or $\tt S^-$, based on the following criteria: $t_k = \tt A^*$ if $B$ has both a positive and a negative order-$k$ principal minor, and each order-$k$ principal minor is nonzero. $t_k = \tt A^+$ (respectively, $t_k = \tt A^-$) if each order-$k$ principal minor is positive (respectively, negative). $t_k = \tt N$ if each order-$k$ principal minor is zero. $t_k = \tt S^*$ if $B$ has each a positive, a negative, and a zero order-$k$ principal minor. $t_k = \tt S^+$ (respectively, $t_k = \tt S^-$) if $B$ has both a zero and a nonzero order-$k$ principal minor, and each nonzero order-$k$ principal minor is positive (respectively, negative). In this talk, results regarding the attainability of epr- and sepr-sequences are discussed. In particular, results forbidding certain subsequences from appearing in the sepr-sequence of a Hermitian matrix are presented. The notion of a $\textit{nonnegative}$ and $\textit{nonpositive}$ subsequence is introduced, which leads to a connection with positive semidefinite matrices

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We consider upper Hessenberg unitary-plus-rank-one matrices, that is, matrices of the form $A = \tilde{U} + \tilde{x}\tilde{y}^{T}$, where $\tilde{U}$ is unitary, and $A$ is upper Hessenberg. This includes the class of Frobenius companion matrices, so methods for this type of matrix can be applied to the problem of finding the zeros of a polynomial. The unitary-plus-rank-one structure is preserved by any method that performs unitary similarity transformations, including Francis's implicitly-shifted $QR$ algorithm. We present a new implementation of Francis's algorithm that acts on a data structure that stores the matrix in $O(n)$ space and performs each iteration in $O(n)$ time. This is joint work with Jared Aurentz, Thomas Mach, and Raf Vandebril. Ours is not the first fast algorithm that has been proposed for this problem, but it is the first that has been shown to be backward stable, and it is faster than other fast algorithms that have been proposed previously.

In 2015, Armario conjectured that there exists a skew-symmetric EW matrix of order $4t+2$ if and only if there exists a tournament matrix $A$ with characteristic polynomial \[ \chi_A(x) = \left ( x^3 - (2t-1)x^2 - t(4t-1) \right ) \left ( x^2+x+t \right )^{2t-1}. \] In this talk, we prove Armario's conjecture . This is based on joint work with Gary Greaves.

According to a long standing conjecture (sometimes called the Perfect-Mirsky conjecture), the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of all regular $k$-gons with $2 \leq k\leq n$ and anchored at 1 in the closed unit disc. It is easy to verify this fact for $n =2$ and $n=3$. But, for $n\geq  4$, it was an open question at least since 1965. Mashreghi-Rivard (2007) showed that the conjecture is wrong for $n = 5$. Then Levick-Pereira-Kribs (2014) added to the mystery by showing that the conjecture is true for $n=4$. For $n \geq 6$, the loci of eigenvalues is unknown. They also came up with a new formulation of the conjecture.

An $n$-by-$m$ Cauchon diagram $C$ is an $n$-by-$m$ grid consisting of $n∙m$ squares colored black and white, where each black square has the property that every square to its left (in the same row) or every square above it (in the same column) is black. Let $A=(a_{ij})$ be an $n$-by-$m$ matrix and $C$ an $n$-by-$m$ Cauchon diagram. Then we say that $A$ is a Cauchon matrix associated with the Cauchon diagram $C$ if for all $(i,j) \in \{1,…,n\} \times \{1,…,m\}$, we have $a_{ij}=0$ if and only if the corresponding square $(i,j)$ in $C$ is black. In this talk, we present a novel method for the determination of the rank of a matrix A and for checking a set of its consecutive row (or column) vectors for linear independence provided that the resulting matrix $\tilde{A}$ of the application of the condensed form of the Cauchon algorithm, see e.g., [2], is a Cauchon matrix. This method is also linked to the elementary bidiagonal factorization of a matrix under certain conditions [1]. This is joint work with Khawla Al Muhtaseb and Ayed Abdel Ghani (Palestine Polytechnic University, Hebron, Palestine), Shaun M. Fallat (University of Regina, Regina, Canada), and Juergen Garloff (University of Applied Sciences / HTWG Konstanz, and University of Konstanz, Konstanz, Germany). References: [1] M. Adm, K. Al Muhtaseb, A. Abedel Ghani, S. Fallat, and J. Garloff, A novel method for determining the rank of a matrix with application to bidiagonal factorization, submitted. [2] M. Adm and J. Garloff, Improved tests and characterizations of totally nonnegative matrices, Electron. J. Linear Algebra, 27, 588-610, 2014.

New and old results will be presented on the Envelope, $E(A)$, which is a bounded region in the complex plane that contains the eigenvalues of a complex matrix $A$. $E(A)$ is the intersection of an infinite number of regions defined by elliptic curves. As such, $E(A)$ resembles and is contained in the numerical range of $A$, which is the intersection of an infinite number of half-planes. The Envelope, however, can be much smaller than the numerical range, while not being much harder to compute. The talk is based on joint work with Panos Psarrakos, Maria Adam and Katerina Aretaki.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph. Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix. We characterize the graphs that are diagonalizable by the standard Hadamard matrix, showing a direct relationship to cubelike graphs. We give some example constructions illustrating our results. ${\bf Co-author(s):}$ N. Johnston (Mount Allison University), S. Kirkland (University of Manitoba), R. Storey (Brandon University), and X. Zhang (University of Manitoba)

It is well known that a complex zero-nonzero pattern cannot be spectrally arbitrary if its digraph doesn’t have at least two loops and at least one two cycle, or at least three loops. This talk focuses on several other combinatorial conditions on the digraph that preclude it from being spectrally arbitrary. In particular we are sometimes able to reduce the number of unknown entries to be below the threshold of $2n−1$. Furthermore there are several algebraic conditions on the coefficients of the characteristic polynomial that can be exploited to show that a pattern is not spectrally arbitrary over any field. Using Sage we were able to show that no zero-nonzero pattern with $2n−1$ nonzero entries will be spectrally arbitrary over $C$ where $n≤6$. When $n=7$ we find two zero-nonzero patterns that do not satisfy our algebraic conditions precluding them from being spectrally arbitrary.

The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. We investigate Hadamard matrices with few distinct types. Apparently, Hadamard matrices with few distinct types are very rare and have nice combinatorial properties. We show that there exists no Hadamard matrix of order larger than $12$ whose quadruples of rows are all of the same type. We then focus on Hadamard metrics with two distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types. This is a joint work with A. Mohammadian.

This talk concerns the corona type Bezout equation $G(z)X(z)=I_p$, $z$ in $BD$. The function $G$ is the given function and $X$ is the unknown. Both functions are stable rational matrix functions, $G$ is of size $p\times q$ and $X$ of size $q\times p$, and $I_p$ stands for the $p\times p$ identity matrix. Here \emph{stable} means that the poles of the rational matrix functions involved are all outside the closed unit disc, and hence the entries of these matrix functions are $H^\infty$- functions. We shall use state space techniques from mathematical system theory to obtain necessary and sufficient conditions for existence of solutions. Furthermore, we derive an explicit formula for the least squares solution and an explicit description of all solutions, all in terms of a state space realization of the given function $G$. The state space formula for the least squares solution is easy to use in Matlab, and it shows that the McMillan degree of this solution is less than or equal to the McMillan degree of $G$. The talk is based on joint work with Art Frazho (Purdue University) and Andre Ran (VU Amsterdam).

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.