Thursday, August 15, 2019 2:30 pm
-
2:30 pm
EDT (GMT -04:00)
Title: Upsetting Matrices
| Speaker: | Chris Godsil |
| Affiliation: | University of Waterloo |
| Room: | MC 5479 |
Abstract:
If
$A$
and
$P$
are
$n\times
n$
matrices
and
the
entries
of
$B$
are
small,
we
may
view
$A+B$
as a
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
Hermitian and
$t$
is
real,
so-called
Hermitian
pencils.
For
example,
if
$A$
is
a
symmetric
weighted
adjacency
matrix
of
a
graph
$X$, then
the
number
of
non-negative
eigenvalues
of
$A$
is
an
upper
bound
on
the
independence
number
of
$X$.
It
is then
natural
to
consider
what
happens
to
the
eigenvalues
when
we
change
the
weight
of
an
edge. Related
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
theory predicts,
and
only
proving
the
simplest
parts.