


Matrix Analysis
Org: ManDuen Choi (Toronto) and Douglas Farenick (Regina) [PDF]
 TSUYOSHI ANDO, Hokkaido University, Sapporo, Japan
Indefinite Contractions
[PDF] 
Given a matrix A, define the triple (p_{}(A),p_{0}(A), p_{+}(A))
as its inertia with respect to the unit circle where p_{}(A)
(resp. p_{+}(A)) is the number of eigenvalues of A inside
(resp. outside) the unit disc while p_{0}(A) is the number of
eigenvalues on the unit circle.
An invertible Hermitian matrix H gives rise to an (indefinite) inner
product. A matrix A is called an H(strict) contraction (or
Hcontractive) if H > A^{*} HA. The most interesing case is that
H is an involution, H^{2} = I. We use J for H in such a case.
It is well known that a matrix A is Hcontractive for suitable H
if and only if p_{0}(A) = 0. Though H is not determined uniquely
by A (even up to positive scalar multiple), p_{}(A) must coincide
with the number of positive eigenvalues of H.
If A is Jcontractive, so is A^{*} and hence A^{*}A. Therefore
A and its modulus A º (A^{*}A)^{1/2} have the same inertia.
But this property does not seem sufficient to guarantee
Jcontractivity of A for a suitable involution J. Other
necessary conditions are presented.
If Hcontractivity of a matrix A always guarantees that of its
adjoint A^{*} then H is necessarily a scalar multiple of an
involution.
A characterization is given for a set of matrices coincides with the
set of Hcontractions for (unknown) H.
 PAUL BINDING, University of Calgary
A generalized eigenvalue problem in the maxalgebra
[PDF] 
We consider the generalized eigenvalue problem
AÄx = lBÄx, x ³ 0, x ¹ 0, 

where A and B are (entrywise) nonnegative n×n matrices,
and the "max" product Ä satisfies
(AÄx)_{i} : = 
n max
m=1

a_{im} x_{m}. 

The case B=I has been studied by several authors, and for
irreducible (e.g., positive) A there is exactly one
eigenvalue l in the above "max" sense.
The generalized problem is different, and for example neither
existence nor uniqueness of eigenvalues is guaranteed, even for
2×2 positive matrices A and B. This case can be analysed
by graphical methods, but for general n, degree theory turns out to
be a more useful tool.
 WAISHUN CHEUNG, The University of Hong Kong, Hong Kong
A matrix approach to the study of generalised derivatives of
polynomials
[PDF] 
Consider a polynomial p and a generalised derivative q of p. We
construct two matrices whose characteristic polynomials are p and
q respectively, and use the properties of the two matrices to study
the relations between the zeros of p and the zeros of q. This
approach is a generalization of the concept of differentiators
recently revived by R. Pereira.
 MANDUEN CHOI, University of Toronto, Department of Mathematics, Toronto,
Ontario M5S 2E4
Completely Positive Linear Maps
[PDF] 
This is a survey talk on the structure of completely positive linear
maps, which serve as the natural morphisms in the category of matrix
algebras (or operator algebras). It has been well known (for more
than 30 years in electrical circuit theory), that completely positive
linear maps, rather than positive linear maps, are useful connections
for nport networks. Incredibly, many new features of quantum
operations (in recent quantum computing theory) turn out to be old and
new structure problems of completely positive linear maps.
 SHAUN FALLAT, Department of Mathematics & Statistics, University of
Regina, Regina, SK S4S 0A2
Compressions of totally positive matrices
[PDF] 
A matrix is called totally positive if all of its minors are positive.
If a matrix A is partitioned as A = (A_{ij}), i,j = 1,2,...,k,
in which each block A_{ij} is n ×n, then the k ×k
compressed matrix is given by (detA_{ij}). It is wellknown that
if A is positive semidefinite, then the compressed matrix is also
positive semidefinite and that the determinant of the compressed matrix
is larger than detA. For a totally positive matrix A, we show
that the compressed matrix is also totally positive and we verify that
the determinant of the compressed matrix exceeds detA when
k=2,3. An extension that allows for overlapping blocks is also
presented when k=2,3. For k ³ 4 we verify, by example, that
the k ×k compressed matrix of a totally positive matrix need
not be totally positive.
 DOUG FARENICK, University of Regina
Pure matrix states on operator systems
[PDF] 
An operator system is a complex matricially ordered vector space that
is completely order isomorphic to a unital selfadjoint subspace of a
unital C^{*}algebra. A matrix state on an operator system V is a
unital completely positive linear map of V into a full matrix
algebra. Pure matrix states will be discussed, and a new and somewhat
simplified proof of a KreinMilmantype theorem of Webster and
Winkler will be mentioned. If V is 3dimensional, then the matrix
state space of V is matrixaffinely homeomorphic to the matricial
range of some Hilbert space operator. With the aid of this
representation, pure matrix states on 3dimensional operator systems
are examinedand in some cases completely determined.
 PETER GIBSON, York University, 4700 Keele Street, Toronto, ON M3J 1P3
Fast iterative Gaussian quadrature
[PDF] 
We construct a family F of probability distributions on
the real line for which iterated Gaussian quadrature, where the number
of nodes is approximately doubled at each iteration, is
computationally more efficient than usual. For each da in
F, the 2n+1point Gauss rule reuses all the nodes of
the npoint Gauss rule, the 2(2n+1)+1point rule reuses the nodes
of the 2n+1point rule, and so on indefinitely. We show that it is
possible to construct a distribution of this type for an essentially
arbitrary sequence of nodes. This implies, for example, the existence
of a distribution supported on [1,1] whose npoint Gauss rules
have evenly spaced nodes, and (equivalently) whose orthogonal
polynomials of degree n have evenly spaced zeros, for an unbounded
sequence of indices n.
In addition we give an explicit construction for the subclass
O F of F for which iteration of
Gaussian quadrature reuses, not only the nodes, but also the weights
at each step. The classical distribution Ö{1x^{2}} dx is
derived as a particular example.
 RONALD HAYNES, Acadia University
Inverse Positivity of Perturbed MMatrices
[PDF] 
A well known property of an Mmatrix A is that the inverse is
elementwise nonnegative, which we write as A^{1} ³ 0. In this
paper we consider diagonal perturbations of Mmatrices and obtain
bounds on the perturbations so that the nonnegative inverse persists.
 STEVE KIRKLAND, University of Regina, Regina, Saskatchewan
Conditioning of the Entries in Google's PageRank Vector
[PDF] 
The internet search engine Google approaches the problem of ranking
web pages by computing an estimate of the Perron vector of a certain
nonnegative matrix associated with the world wide web. That Perron
vector, known as the PageRank vector, is then used to rank the
importance of the corresponding web pages. In this talk, we discuss
the conditioning of the entries in the PageRank vector, and our
results in turn lead to error bounds for Google's computed estimate of
that vector. Our techniques involve nonnegative matrices, generalized
inverses, and combinatorial considerations.
 IVO KLEMES, McGill University, 805 Sherbrooke St. W. # 1005, Montreal,
QC H3A 2K6
Alexandrov's inequality and some conjectures on Toeplitz
matrices
[PDF] 
We prove the inequality det(QQ^{*}) £ det(RR^{*}) where Q is a
certain Toeplitz matrix associated with the Dirichlet kernel on the
unit circle, and R is any matrix of the same size as Q whose
entries have modulus ³ 1 on the support of the entries of Q,
zero otherwise. This is a special case of a similar problem for all
of the elementary symmetric functions; S_{k}(QQ^{*}) £ S_{k}(RR^{*}), in
the case when R is Toeplitz. The general case is still open. The
proof uses Alexandrov's inequality on the polarized determinant (also
known as the "mixed discriminant"). There are some connections with
totally unimodular matrices, counting of bases modulo p, and
BazinReiszPicquet determinant identities.
 DAVID KRIBS, University of Guelph and Institute for Quantum Computing
Passive Quantum Error Correction
[PDF] 
In this talk, I will discuss some of my recent work on error
correction problems in quantum computing. I'll focus on the
fundamental passive technique, which involves "decoherencefree
subspaces" and "noiseless subsystems". In this method, symmetries
within the noise of quantum channels are used to encode initial states
in sectors of the system Hilbert space that are immune to the errors
of the channel. The underlying mathematics involves completely
positive maps and operator algebras.
 PETER LANCASTER, University of Calgary
The Boundary of the Pseudospectrum
[PDF] 
This work began with the design of an algorithm (work with
P. Psarrakos) to trace the boundary of the pseudo spectrum for matrix
poynomials. Here, the objective is to describe the nature of the
boundary with respect to diferentiability, multiple points, and
perturbations.
This work is in collaboration with P. Psarrakos and L. Boulton.
 CHIKWONG LI, College of William and Mary, Mathematics Department,
Williamsburg, VA 23185, USA
Spectrum preserving maps
[PDF] 
We will discuss recent results on spectrum preserving maps on matrices
and operators, and related problems.
 MATJAZ OMLADIC, University of Ljubljana, Dept. of Math., Jadranska 19,
SI1000 Ljubljana, Slovenia
Semigroups of matrices satisfying commutator equations
[PDF] 
Some recently obtained reducibility results for matrix semigroups
satisfying certain polynomial relations such as commutator relations
will be discussed. For example, we will consider semigroups such that
there is a matrix (not necessarily a member of the semigroup) with the
property that the commutator with every member of the semigroup is
nilpotent.
 RAJESH PEREIRA, University of Saskatchewan, Saskatoon, SK
Majorization Relations in the Geometry of Polynomials
[PDF] 
Majorization (in the sense of HardyLittlewoodPolya and
MarshallOlkin) is an important concept in matrix analysis and the
theory of inequalities. In this talk, we show how matrix analysis can
be used to derive majorization relations between the zeros and the
critical points of polynomials. We will examine both recent work in
this area as well as applications to classical problems in the
geometry of polynomials such as the spans of hyperbolic polynomials
and Mahler measure inequalities.
 HEYDAR RADJAVI, University of Waterloo, Waterloo, Ontario
An Approximate Version of Specht's Theorem
[PDF] 
Specht's theorem states that two complex matrices are unitarily
equivalent if and only if all traces of words in two noncommutative
letters applied to the pairs (A,A^{*}) and (B,B^{*}) coincide. In
joint work with Laurent Marcoux and Mitja Mastnak, we study an
approximate version of this trace hypothesis and obtain conditions
that allow us to extend this theorem to simultaneous similarity or
unitary equivalence of families of operators.
 PETER ROSENTHAL, University of Toronto, Toronto, ON
Equations such as AX + YB = Z in Matrices and Operators
[PDF] 
Let A and B be square matrices of the same size. It is easily
seen that the mapping that sends each pair (X,Y) of square matrices
into the matrix AX + YB is onto if and only if at least one of A
and B is invertible. The analogue is established for operators on
Hilbert space, and partial results are obtained for more general
operator equations in several variables.
This is joint work with Don Hadwin and Eric Nordgren.
 PETER SEMRL, University of Ljubljana, Dept. of Math., Jadranska 19,
SI1000 Ljubljana
Maps on matrix spaces
[PDF] 
Some recently obtained structural results for maps on matrix algebras
(or their subsets) having certain algebraic or preserving properties
will be presented. For example, we will consider maps on the set of
idempotents preserving usual partial order or orthogonality.
Connections with physics and geometry will be disscused.
 MICHAEL TSATSOMEROS, Washington State University, Mathematics, Pullman, WA
991643113, USA
The Principal Minor Assignment Problem
[PDF] 
We will consider the following inverse problem:
[PMAP] Find, if possible, an n×n matrix A having prescribed
principal minors.
We will discuss conditions for the solvability of PMAP, an algorithm
to compute a solution, and consequences in classical problems.

