generalized Bézout theorem on matrices

generalized Bézout theorem on matrices

Generalized Bézout theorem 1

.

Let

M

⁢

[

x

]

be an arbitrary

matrix polynomial

of order

n

and

A

a

square matrix

of the same order. Then, when the matrix polynomial is divided on the right (left) by the

characteristic polynomial

x

⁢

I

-

A

, the

remainder

is

M

⁢

(

A

)

(

M

^

⁢

(

A

)

).

Proof.

Consider

M

⁢

[

x

]

given by

M

[

x

]

=

M

0

x

m

+

M

1

x

m

-

1

+

⋯

+

M

m

,

(

M

0

≠

0

)

.

(1)

The

polynomial

can also be written as

M

⁢

[

x

]

=

x

m

⁢

M

0

+

x

m

-

1

⁢

M

1

+

⋯

+

M

m

.

(2)

We are now substituting the scalar

argument

(real or complex)

x

by the matrix

A

and therefore (1) and (2) will, in general, be distinct, as the powers of

A

need not be permutable with the

polynomial matrix

coefficients

. So that,

M

⁢

(

A

)

=

M

0

⁢

A

m

+

M

1

⁢

A

m

-

1

+

⋯

+

M

m

and

M

^

⁢

(

A

)

=

A

m

⁢

M

0

+

A

m

-

1

⁢

M

1

+

⋯

+

M

m

,

calling

M

⁢

(

A

)

(

M

^

⁢

(

A

)

) the right (left) value of

M

⁢

[

x

]

on substitution of

A

for

x

.

If we divide

M

⁢

[

x

]

by the

binomial

x

⁢

I

-

A

(

I

is the correspondent

identity matrix

), we shall prove that the right (left) remainder

R

(

R

^

) does not depend on

x

. In fact,

M

⁢

[

x

]

=

M

0

⁢

x

m

+

M

1

⁢

x

m

-

1

+

⋯

+

M

m

=

M

0

⁢

x

m

-

1

⁢

(

x

⁢

I

-

A

)

+

(

M

0

⁢

A

+

M

1

)

⁢

x

m

-

1

+

M

2

⁢

x

m

-

2

+

⋯

+

M

m

=

[

M

0

⁢

x

m

-

1

+

(

M

0

⁢

A

+

M

1

)

⁢

x

m

-

2

]

⁢

(

x

⁢

I

-

A

)

+

(

M

0

⁢

A

2

+

M

1

⁢

A

+

M

2

)

⁢

x

m

-

2

+

M

3

⁢

x

m

-

3

+

⋯

+

M

m

=

[

M

0

⁢

x

m

-

1

+

(

M

0

⁢

A

+

M

1

)

⁢

x

m

-

2

+

(

M

0

⁢

A

2

+

M

1

⁢

A

+

M

2

)

⁢

x

m

-

3

]

⁢

(

x

⁢

I

-

A

)

+

(

M

0

⁢

A

3

+

M

1

⁢

A

2

+

M

2

⁢

A

+

M

3

)

⁢

x

m

-

3

+

M

4

⁢

x

m

-

4

+

⋯

+

M

m

=

[

M

0

x

m

-

1

+

(

M

0

A

+

M

1

)

x

m

-

2

+

(

M

0

A

2

+

M

1

A

+

M

2

)

x

m

-

3

+

⋯

+

(

M

0

A

m

-

1

+

M

1

A

m

-

2

+

⋯

+

M

m

-

1

)

]

(

x

I

-

A

)

+

M

0

A

m

+

M

1

A

m

-

1

+

⋯

+

M

m

,

whence we have found that

R

=

M

0

⁢

A

m

+

M

1

⁢

A

m

-

1

+

⋯

+

M

m

≡

M

⁢

(

A

)

,

and analogously that

R

^

=

A

m

⁢

M

0

+

A

m

-

1

⁢

M

1

+

⋯

+

M

m

≡

M

^

⁢

(

A

)

,

which proves the theorem.
∎

From this theorem we have the following

Corollary 1

.

A polynomial

M

⁢

[

x

]

is

divisible

by the characteristic polynomial

x

⁢

I

-

A

on the right (left) without remainder
iff

M

⁢

(

A

)

=

0

(

M

^

⁢

(

A

)

=

0

).

Title

generalized Bézout theorem on matrices

Canonical name

GeneralizedBezoutTheoremOnMatrices

Date of creation

2013-03-22 17:43:35

Last modified on

2013-03-22 17:43:35

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

8

Author

perucho (2192)

Entry type

Theorem

Classification

msc 15-01

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