Thursday, August 15, 2019 2:30 pm
-
2:30 pm
EDT (GMT -04:00)
Title: Upsetting Matrices
| Speaker: | Chris Godsil |
| Affiliation: | University of À¶Ý®ÊÓƵ |
| Room: | MC 5479 |
Abstract:Ìý
If
$A$
and
$P$
are
$n\times
n$
matrices
and
the
entries
of
$B$
are
small,
we
may
view
$A+B$
asÌýa
perturbation
of
$A$,
and
expect
that
the
spectral
properties
of
$A+B$
should
be
related
to
those
of
$A$.In
our
case
we
are
interested
in
linear
combinations
of
the
form
$A+tB$,
where
$A$
and
$B$
are
HermitianÌýand
$t$
is
real,
so-called
Hermitian
pencils.
For
example,
if
$A$
is
a
symmetric
weighted
adjacency
matrix
of
a
graph
$X$,Ìýthen
the
number
of
non-negative
eigenvalues
of
$A$
is
an
upper
bound
on
the
independence
number
of
$X$.
It
isÌýthen
natural
to
consider
what
happens
to
the
eigenvalues
when
we
change
the
weight
of
an
edge.ÌýRelated
problems
arise
in
the
analysis
of
search
algorithms
based
on
continuous
quantum
walks.
I
will
provide
an
introduction
to
perturbation
theory
of
Hermitian
pencils,
focussing
on
what
the
theoryÌýpredicts,
and
only
proving
the
simplest
parts.