




Matrix Analysis / Analyse matricielle (Judith McDonald, Organizer)
 ROD EDWARDS, University of Victoria, Victoria, British Columbia V8W 3P4
Common invariant proper cones and ray dynamics

Certain problems in dynamics of networks give rise to dynamical systems
in which matrix operators act on the space of rays in R^{n}.
One important problem is to find conditions on a pair (or set) of real
matrices such that rays become localized under the action of `words'
composed of multiple products of these matrices, and to find regions of
R^{n} where they are confined. If eigenvalues of the matrices
are real, for example, one may look for conditions giving a common
invariant proper cone, K, for the two matrices. In general, this
problem appears to be hard, but we present some partial results
including a sufficient condition in the case of R^{3}.
Negative dominant eigenvalues may still allow invariant sets of the
form KÈ(K), where K is a proper cone, and dominant complex
eigenvalues may allow other forms of invariant set.
 SHAUN FALLAT, Department of Mathematics and Statistics, University of
Regina, Regina, Saskatchewan S4S 0A2
Eigenvalues of products of matrices and submatrices
in certain positivity classes

If A and B are nbyn, positive semidefinite Hermitian
matrices, then
l_{min}(AB) £ l_{min}(A[a]B[a]), 

for any f ¹ a Í {1,2,¼,n}. A certain
converse is given, as well as analogs for the product of several
Mmatrices and totally nonnegative matrices. However, analogs for
l_{max} (other than the case of totally nonnegative matrices)
and for singular values of general matrices, etc., fail.
 CHUNHUA GUO, University of Regina, Regina, Saskatchewan
Iterative methods for a nonlinear matrix equation

We are concerned with the numerical computation of the maximal
Hermitian positive definite solution of the matrix equation
X+A^{*}X^{1}A=Q, where Q is Hermitian positive definite. Three
iterative methods are discussed: a simple fixed point iteration, the
Newton method, and a method based on cyclic reduction.
 ALLEN HERMAN, University of Regina, Regina, Saskatchewan
A maximal eigenvalue conjecture equivalent to permanental dominance

Using concepts from the character theory of finite groups, a
conjecture about maximal eigenvalues will be presented that is
equivalent to Lieb's permanental dominance conjecture.
 HADI KHARAGHANI, University of Lethbridge, Lethbridge, Alberta
wshift matrices and applications

Let C_{n}=\prec w\succ be a cyclic group of order n. The matrix
UW, where U is the circulant matrix of order n with first
row (0,1,0,¼,0) and W = diag(w,1,¼,1), is called
an wshift matrix. Let q be a prime power and n a
positive integer with [(q1)/(n)] being an even (respectively odd)
integer. Using a cyclic group of wshift matrices, we show that
there is a symmetric (respectively skew) BGW(1+q+¼+q^{2m+1},q^{2m+1},q^{2m+1}q^{2m}) with zero diagonal over the cyclic group
C_{n} of order n, for each nonnegative integer m. Applications
include a large class of new (respectively directed) Strongly Regular
Graphs.
 STEVE KIRKLAND, University of Regina, Regina, Saskatchewan S4S 0A2
A bound on algebraic connectivity for graphs with cutpoints

For a graph G, its Laplacian matrix can be written as L=DA,
where A is the (0,1) adjacency matrix of G, and D is the
diagonal matrix of vertex degrees. The algebraic connectivity of G
is the second smallest eigenvalue of L, and as the name suggests, it
seems to provide an algebraic measure of the connectedness of G. A
vertex of G is a cutpoint if its deletion disconnects the graph. In
this talk, we consider graphs on n vertices with k cutpoints,
provide a tight upper bound on the algebraic connectivity in terms of
n and k, and characterize the graphs yielding equality in the
bound.
 CHIKWONG LI, William and Mary, International Linear Algebra Society
Speaker, Williamsburg, Virginia 23187, USA
Linear Preserver Problems: A gentle introduction and some
recent results

Linear preserver problems concern the characterizations of linear
transformations on matrix or operator spaces that have some special
properties. In this talk, a gentle introduction of the subject
including history, motivations of study, and some current research will
be presented. Recent results obtained by the speaker and his
collaborators will also be described.
 GORDON MACDONALD, University of Prince Edward Island, Charlottetown, Prince Edward
Island
Ntransitivity and the complementation property

The structure of (minimal) ntransitive rankn semigroups of k×k matrices of is investigated by considering an equivalent
problem regarding certain families of (kn)dimensional subspaces of
the underlying vector space.
 JOHN MAROULAS, National Technical University of Athens
On compressions of normal matrices

Let A be a normal matrix and consider the polygon NR[A]={x^{*}Ax:x=1 }. If u^{*}Au Î NR[A], a projector matrix
P_{u} is defined such that NR[P^{*}_{u}AP_{u}] is supported by all or some edges of the polygon. The
inverse problem for a matrix G, and the dilation of NR[G] to a
circumscribed hexagon are also investigated.
 CARL MEYER, Department of Mathematics, North Carolina State University,
SIAM Speaker
Replacing the SVD in LSI and information retrieval
applications

Latent semantic indexing (LSI) is a vector space technique that
utilizes the SVD to process queries in an information retrieval
system such as an internet search engine. Some of the bottlenecks
associated with the SVD approach include storage considerations,
difficulties in updating and downdating the underlying data base, and
speed vs. reliability issues. After reviewing some of these
basic problems, some different vector space approaches to LSI that
do not involve an SVD decomposition will be presented and
discussed.
 DALE OLESKY, Department of Computer Science, University of Victoria, Victoria,
British Columbia
Successivelyordered elementary bidiagonal factorization

Let D be a diagonal matrix and E_{ij} denote the nbyn matrix
with a 1 in entry (i,j) and 0 in every other entry. An
nbyn matrix A has a successivelyordered elementary bidiagonal
(SEB) factorization if it can be factored as
A = 
æ è


n1 Õ
k=1


k+1 Õ
j=n

L_{j}(s_{jk}) 
ö ø

D 
æ è


1 Õ
k=n1


n Õ
j=k+1

U_{j}(t_{kj}) 
ö ø

, 

in which L_{j}(s_{jk})=I+s_{jk}E_{j,j1} and U_{j} (t_{kj}) = I+t_{kj}E_{j1,j} for some scalars s_{jk}, t_{kj}. Note that some
of the parameters s_{jk},t_{kj} may be zero, and the order of the
bidiagonal factors is fixed. If this factorization corresponds to
reduction of A to D via successive row/column operations in the
specified order, it is called an elimination SEB
factorization. New rank conditions are formulated that are proved to
be necessary and sufficient for matrix A to have such a
factorization. These conditions are related to known but more
restrictive properties that ensure a bidiagonal factorization as above,
but with all parameters s_{jk}, t_{kj} nonzero.
 P.N. SHIVAKUMAR, University of Manitoba, Winnipeg, Manitoba R3T 3N2
Some nonsingular matrices and their applications

Nonsingular matrices with various structures involving certain sign
patterns, diagonal dominance, tridiagonal, etc., are presented.
Applications include, extensions to the infinite case, Digital Circuit
Dynamics, conformal mapping of doubly connected regions and double
points of a Mathieu equation.
 MICHAEL TSATSOMEROS, University of Regina, Department of Mathematics and Statistics,
Regina, Saskatchewan S4S 0A2
PerronFrobenius type results on the numerical range

We present results connecting the shape of the numerical range to
intrinsic properties of a nonnegative matrix A. These results are to
a large extent analogous to the PerronFrobenius theory, especially as
it pertains to irreducibility and cyclicity in the combinatorial sense.
Special attention is given to circular and elliptic numerical ranges.
 PETER ZIZLER, Mount Royal College, Calgary, Alberta
Norms of sampling operators

Let h(z) be an essentially bounded complex valued function on the
unit circle T={z z=1}. Consider the
corresponding Laurent operator L_{h}=(h_{nm})_{n,m Î Z},
where h_{n} is the nth Fourier coefficient of h(z),
h_{n}=ò_{T} h(z)z^{n} dz.
Let us consider an operator S_{h}(p,q), which we shall call a
sampling operator, defined as S_{h}(p,q)=(h_{pnqm})_{n,m Î Z}, where p,q Î N={1,2,... }. These
operators are obtained from L_{h} be ``keeping'' every qth column
and every pth row in the biinfinite matrix L_{h}. In our paper, we
find an upper and lower bound for the norm of the operator S_{h}(p,q).

