Embree
– draft –
23 February 2012

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1.8.
Nondiagonalizable Matrices: The Jordan Form
35
Characteristic Polynomial and Minimal Polynomial
Suppose the matrix
A
∈
n
×
n
has the Jordan structure detailed in
Theorem 1.7. The
characteristic polynomial
of
A
is given by
p
A
(
z
) =
p
j
=1
(
z
−
λ
j
)
a
j
,
a degree-
n
polynomial; the
minimal polynomial
of
A
is
m
A
(
z
) =
p
j
=1
(
z
−
λ
j
)
d
j
.
Since the algebraic multiplicity can never exceed the index of an eigen-
value (the size of the largest Jordan block), we see that
m
A
is a polynomial
of degree no greater than
n
. If
A
is not derogatory, then
p
A
=
m
A
; if
A
is derogatory, then the degree of
m
A
is strictly less than the degree of
p
A
,
and
m
A
divides
p
A
. Since
(
J
j
−
λ
j
I
)
d
j
=
N
d
j
j
=
0
,
notice that
p
A
(
J
j
) =
m
A
(
J
j
) =
0
,
j
= 1
, . . . , p,
and so
p
A
(
A
) =
m
A
(
J
j
) =
0
.
Thus, both the characteristic polynomial and minimal polynomials
annihi-
late
A
. In fact,
m
A
is the lowest degree polynomial that annihilates
A
. The
first fact, proved in the 2
×
2 and 3
×
3 case by
Cayley
[Cay58], is known
as the
Cayley–Hamilton Theorem
Cayley–Hamilton Theorem
Theorem 1.8.
The characteristic polynomial
p
A
of
A
∈
n
×
n
annihi-
lates
A
:
p
A
(
A
) =
0
.
Embree
– draft –
23 February 2012