Title: Investigation of Crouzeix's Conjecture via Nonsmooth Optimization
Speaker: | Michael Overton |
Affiliation: | Courant Institute of Mathematical Science, New York University |
Room: | MC 5501 |
Abstract:
Crouzeix's
conjecture
is
among
the
most
intriguing
developments
in
matrix
theory
in
recent
years.
Made
in
2004
by
Michel
Crouzeix,
it
postulates
that,
for
any
polynomial
p
and
any
matrix
A,
||p(A)||
<=
2
max(|p(z)|:
z
in
W(A)),
where
the
norm
is
the
2-norm
and
W(A)
is
the
field
of
values
(numerical
range)
of
A,
that
is
the
set
of
points
attained
by
v*Av
for
some
vector
v
of
unit
length.
Remarkably,
Crouzeix
proved
in
2007
that
the
inequality
above
holds
if
2
is
replaced
by
11.08.
Furthermore,
it
is
known
that
the
conjecture
holds
in
a
number
of
special
cases,
including
n=2.
We
use
nonsmooth
optimization
to
investigate
the
conjecture
numerically
by
attempting
to
minimize
the
"Crouzeix
ratio",
defined as
the
quotient
with
numerator
the
right-hand
side
and
denominator
the
left-hand
side
of
the
conjectured
inequality.
We
present
numerical
results
that
lead
to
some
theorems
and
further
conjectures,
including
variational
analysis
of
the
Crouzeix
ratio
at
conjectured
global
minimizers.
All
the
computations
strongly
support
the
truth
of
Crouzeix's
conjecture.
This
is
joint
work
with
Anne
Greenbaum
and
Adrian
Lewis.