Let
V
be a finite dimensional vector space. What is the minimal
polynomial for the identity operator on
V
? What is the minimal polynomial for
the zero operator?
The minimal polynomial for the identity operator is
p
(
x
) =
x

1
.
It is monic, of degree
1
and it annihilates the identity operator.
The minimal
polynomial for the zero operator is
p
(
x
) =
x
.
Let
A
be the
4
×
4
real matrix
A
=
1
1
0
0

1

1
0
0

2

2
2
1
1
1

1
0
Show that the characteristic polynomial for
A
is
x
2
(
x

1)
2
and that it is also the
minimal polynomial.
Calculating
det(
xI

A
)
we get
x
2
(
x

1)
2
.
The minimal polynomial
should have the same degree
1
factors, i.e.
x
and
(
x

1)
. Calculating the remain
ing possibilities we get:
A
(
A

I
) = 0
,
A
2
(
A

I
) = 0
,
A
(
A

1)
2
= 0
. Therefore the
minimal polynomial is the characteristic polynomial.
Is the matrix
A
of Exercise 3 similar over the field of complex
numbers to a diagonal matrix?
One can easily check that the matrices
A
and
A

I
have rank
3
.
Hence,
A
has exactly two eigenvectors: one with eigenvalue
0
, and the other with
eigenvalue
1
. So
A
does not have a basis of eigenvectors, and thus is not similar
to a diagonal matrix over the complex field.
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 Spring '10
 aa
 Derivative, Vector Space, minimal polynomial

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