5Eigenvalues and eigenvectors

IA Vectors and Matrices

5.7 Cayley-Hamilton Theorem

Theorem

(Cayley-Hamilton theorem)

.

Every

n × n

complex matrix satisfies

its own characteristic equation.

Proof.

We will only prove for diagonalizable matrices here. So suppose for our

matrix

A

, there is some

P

such that

D

=

diag

(

λ

1

, λ

2

, ··· , λ

n

) =

P

−1

AP

. Note

that

D

i

= (P

−1

AP )(P

−1

AP ) ···(P

−1

AP ) = P

−1

A

i

P.

Hence

p

D

(D) = p

D

(P

−1

AP ) = P

−1

[p

D

(A)]P.

Since similar matrices have the same characteristic polynomial. So

p

A

(D) = P

−1

[p

A

(A)]P.

However, we also know that D

i

= diag(λ

i

1

, λ

i

2

, ···λ

i

n

). So

p

A

(D) = diag(p

A

(λ

1

), p

A

(λ

2

), ··· , p

A

(λ

n

)) = diag(0, 0, ··· , 0)

since the eigenvalues are roots of

p

A

(

λ

) = 0. So 0 =

p

A

(

D

) =

P

−1

p

A

(

A

)

P

and

thus p

A

(A) = 0.

There are a few things to note.

(i)

If

A

−1

exists, then

A

−1

p

A

(

A

) =

A

−1

(

c

0

+

c

1

A

+

c

2

A

2

+

···

+

c

n

A

n

) = 0.

So

c

0

A

−1

+

c

1

+

c

2

A

+

···

+

c

n

A

n−1

. Since

A

−1

exists,

c

0

=

±det A 6

= 0.

So

A

−1

=

−1

c

0

(c

1

+ c

2

A + ··· + c

n

A

n−1

).

So we can calculate A

−1

from positive powers of A.

(ii) We can define matrix exponentiation by

e

A

= I + A +

1

2!

A

2

+ ··· +

1

n!

A

n

+ ··· .

It is a fact that this always converges.

If

A

is diagonalizable with

P

with

D

=

P

−1

AP

=

diag

(

λ

1

, λ

2

, ··· , λ

n

),

then

P

−1

e

A

P = P

−1

IP + P

−1

AP +

1

2!

P

−1

A

2

P + ···

= I + D +

1

2!

D

2

+ ···

= diag(e

λ

1

, e

λ

2

, ···e

λ

n

)

So

e

A

= P [diag(e

λ

1

, e

λ

2

, ··· , e

λ

n

)]P

−1

.

(iii)

For 2

×

2 matrices which are similar to

B

=

λ 1

0 λ

We see that the

characteristic polynomial

p

B

(

z

) =

det

(

B −zI

) = (

λ −z

)

2

. Then

p

B

(

B

) =

(λI − B)

2

=

0 −1

0 0

2

=

0 0

0 0

.

Since we have proved for the diagonalizable matrices above, we now know

that any 2 × 2 matrix satisfies Cayley-Hamilton theorem.

In IB Linear Algebra, we will prove the Cayley Hamilton theorem properly for

all matrices without assuming diagonalizability.