Matrices
A
(
t
)
depending on a Parameter
t
Math. 412, Fall 1995
Jerry L. Kazdan
If a square matrix
A
(
t
)
depends smoothly on a parameter
t
, are its eigenvalues and eigen
vectors also smooth functions of
t
? The answer is “yes” most of the time, but not always.
This story, while old, is interesting and elementary — and deserves to be better known.
One can also ask the same question for objects such as the Schr¨odinger operator whose
potential depends on a parameter, where much of current understanding arose.
Warmup Exercise
Given a polynomial
p
(
x
,
t
) =
x
n
+
a
n

1
(
t
)
x
n

1
+
···
+
a
1
(
t
)
x
+
a
0
(
t
)
whose coef±cients
depend smoothly on a parameter
t
. Assume at
t
=
0 the number
x
=
c
is a simple root
of this polynomial,
p
(
c
,
0
) =
0. Show that for all
t
suf±ciently near 0 there is a unique
root
x
(
t
)
with
x
(
0
) =
c
that depends smoothly on
t
. Moreover, if
p
(
x
,
t
)
is a real analytic
function of
t
, that is, it has a convergent power series expansion in
t
near
t
=
0, then so
does
x
(
t
)
.
S
OLUTION
: Given that
p
(
c
,
0
) =
0 we want to solve
p
(
x
,
t
) =
0 for
x
(
t
)
with
x
(
0
) =
c
.
The assertions are immediate from the implicit function theorem.
Since
x
(
0
) =
c
is a
simple zero of
p
(
x
,
0
) =
0, then
p
(
x
,
0
) = (
x

c
)
g
(
x
)
, where
g
(
c
)
6
=
0. Thus the derivative
p
x
(
c
,
0
)
6
=
0.
The example
p
(
x
,
t
)
:
=
x
3

t
=
0, so
x
(
t
) =
t
1
/
3
, shows
x
(
t
)
may not be a smooth function
at a multiple root. In this case the best one can get is a Puiseux expansion in fractional
powers of
t
(see [Kn, ¶15]).
The Generic Case: a simple eigenvalue
In the following, let
λ
be an eigenvalue and
X
a corresponding eigenvector of a matrix
A
.
We say
λ
is a
simple eigenvalue
if
λ
is a simple root of the characteristic polynomial. We
will use the equivalent version:
if
(
A

λ
)
2
V
=
0
, then V
=
cX for some constant c
. The
point is to eliminate matrices such as the zero 2
×
2 matrix
A
=
0, where
λ
=
0 is a double
eigenvalue and any vector
V
6
=
0 is an eigenvector, as well as the more complicated matrix
A
=
(
0 1
0 0
)
which has
λ
=
0 as an eigenvalue with geometric multiplicity one but algebraic
multiplicity two.
Theorem