On Polynomial Symmetric and Polynomial Skew Symmetric Matrices

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On Polynomial Symmetric and Polynomial Skew Symmetric Matrices

G. Ramesh

1

P. N. Sudha

2

1

Associate Professor

2

Assistant Professor

1,2

Department of Mathematics Engineering

1

Govt Arts College (Auto), Kumbakonam, Tamil Nadu, India

2

Periyar Maniammai University,

Thanjavur, Tamil Nadu, India

Abstract— some of the properties of symmetric and skew symmetric matrices are extended to polynomial symmetric and polynomial skew symmetric matrices. Characterizations of polynomial symmetric and polynomial skew symmetric matrices are also presented.

Key words: Polynomial Symmetric Matrix, Polynomial Skew Symmetric Matrix, AMS Classification: 15A09, 15A15, 15A57.

I. INTRODUCTION

Let

nxn

C

A be a complex matrix. It is symmetric if _A__AT and is skew symmetric is _A_AT_where T

A is the transpose

of

A

.

This symmetric matrix is used in statistical analysis, theory of graphs and networks[1]. Complex symmetric matrices arise in the study of damped vibrations of linear systems, in classical theories of wave propagation in continuous media, and in general relativity.

A matrix A() is said to be a Polynomial matrix if all entries of A() are polynomials. Polynomials and polynomial matrices arise naturally as modeling tools in several areas of applied mathematics, science and engineering, especially in systems theory [2], [7], [8].

In [3] and [4] we have studied about polynomial hermitian, polynomial normal and polynomial unitary matrices and discussed some properties. Also we found determinant value of a polynomial matrix, product of two polynomial matrices.

In this paper we have extended the properties of symmetric and skew symmetric matrices to polynomial symmetric and polynomial skew symmetric matrices.

II. PRELIMINARIES

A. Definition 2.1[6]

The number of linearly independent rows (columns) of a polynomial matrix () mxn()

R

A  is called its normal rank. B. Remark 2.2

The rank of a polynomial matrix A()Rm x n() is not greater than the minimum of the number of its rows and columns. That is rankA()min(m,n).

If the square polynomial matrix is of order n and is of full rank, that is rankA()n,then A()is said to be non-singular. In other words if the determinant of a polynomial matrix A() is nonzero then A() is said to be nonsingular. The rank of a nonsingular square polynomial matrix of order n is equal to n.

C. Example 2.3

Let

   

 

 

 

  

 

2 1

1 1 )

( ₂

A

. 0 1 ) (  

A Det

It is nonsingular. Here this matrix formed by linearly independent elements. Hence rankA(



)2

D. Remark 2.4

1) The rank of the product of two polynomial matrices )

(

A and B()cannot exceed the smallest rank of the

multiplicand matrices. That is if

). , min( )) ( ) ( (

, ) ( )

(

2 1

r r B

A rank

r is B of rank the and r is A of rank

  

 

2) The rank of the sum of two polynomial matrices A()

and B() cannot exceed the sum of ranks of the summand

matrices. That is rankofA()isr1andtherankofB()isr2,

2 1

)) ( ) (

(A B r r

rank      .

3) A polynomial matrix is column reduced if its column leading coefficients matrix has full column rank. It is row reduced if its row leading coefficient matrix has full row rank.

E. Definition 2.5 [6]

Two polynomial matrices ₍__), ₍_₎ mxn₍_₎

C B

A  are called

left or row equivalent if and only if one of them can be obtained from the other as a result of a finite number of elementary operations carried out on its rows.

) ( ) ( )

( L A

B  , where L() is unimodular matrix of order m. (_L₍_₎_₀).

F. Definition 2.6 [6]

Two polynomial matrices A(),B()Cmxn() are called right or column equivalent if and only if one of them can be obtained from the other as a result of a finite number of elementary operations carried out on its columns.

) ( ) ( )

( A P

B  , where P() is unimodular matrix of order n. (_P₍_₎_₀).

G. Definition 2.7 [6]

Two polynomial matrices ₍__), ₍_₎ mxn₍_₎

C B

A  are

equivalent thenB()L()A()P()_,where L()and P() is unimodular matrices of order m and n respectively.

H. Definition 2.8

A polynomial identity matrix is a matrix whose coefficient matrices are identity matrices.

I. Example 2.9

. 1 0

0 1

1 0

0 1

1 0

0 1 )

(



_

                

 

  

(2)

J. Definition 2.10 [6]

A polynomial matrix obtained from the polynomial identity matrix by applying a elementary row or column operation is called an elementary polynomial matrix.

K. Example 2.11

. 1 0

0 1 )

( _

  

 

  

  

A Let

For example

   

 

     

 

 

) 1 ( 0

0 ) 1 ( 4 , 0 1

1 0

  

 _are

elementary polynomial matrices obtained from the polynomial identity matrix

A

(



)

by applying the elementary operations

1 1 2

1 R ,R 4R

R   respectively.

L. Remark 2.12

 Inverse of an elementary polynomial matrix is again a elementary polynomial matrix.

 Any elementary polynomial matrix is non-singular.

M. Example 2.13

  

 

1 0

0 1

1 0

0 1

1 0

0 1 ) (

A A A

Let

 

                 

 

  

. 0 2 1

)

(  2   A

Det Here elementary polynomial

matrix is non-singular.

N. Definition 2.14 [5]

A polynomial matrix

A

(



)

is conjugate normal if .

) ( ) ( ) ( ) (

___ __________

 _ _ _



 A A A

A

P. Example 2.15

Let

















2

1

)

(



A

   

 

   

 



 



1 2 2

2 1

) ( ) ( ) ( ) (

2 4 2

2 2

___ __________

    

  

 A A A

A

O. Example 2.16

Complex polynomial symmetric, complex polynomial skew-symmetric and real polynomial normal matrices are conjugate normal.

III. POLYNOMIAL SYMMETRIC AND POLYNOMIAL SKEW

SYMMETRIC MATRICES

A. Definition 3.1

A square polynomial matrix A()is said to be symmetric if )

(

A = _A₍₎T_{in other words all the coefficient matrices of}

) (

A are symmetric.

B. Theorem 3.2

Let A()and B()be polynomial symmetric matrices of order n. Then

1) A()+ B() is symmetric.

2) A()B()is symmetric if and only if A()B()₌B()A()_.

3) A()B()+ B()A() is symmetric.

4) If A() is symmetric then kA()is symmetric, where F

k _.

C. Proof

1) [A( ) B()]T = A()T+ B()T = A()+ B()

Hence A()+ B() is symmetric.

2) A()B()is symmetric



_[_A₍₎_B₍_)]T

= A()B()



_B₍₎T_A₍₎T

= A()B()



B()A()₌A()B()_.

3)

T

A B B

A( ) ( ) ( ) ( )]

[     

= [A()B()]T [B()A()]T =

T T T

T

B A A

B(



) (



)  (



) (



)

= B()A()+ A()B() = A()B() + B()A()

4) [KA()]T K[A()]T KA().

D. Alternate Proof

Given

n n

A A

A()   2... 

2 1

0 and

n n B B

B B

B()   2... 

2 1 0

are polynomial symmetric matrices, where the coefficient matrices A0,A1,A2,....An_and B0,B1,B2...Bn_{are symmetric}

matrices. That is

T n n T T

A A A A A

A0 0 , 1 1 ,...,  _and

. ,

,...

, ₁ ₁

0 0

T n n T

T

B B B

B B

B   

1) Let us consider

n n

n B

A

B A B A B A



 



) (

... ) ( ) ( ) ( )

( 0 0 1 1

 

     

We know that “A and B are symmetric then A + B is symmetric”. From the result we have

. ) ( .

..., ,... ) ( ,

)

( 0 0 1 1 1 1

0 0

T n n n n

T T

B A B A

B A B A B A B A

  

   



 Hence the

coefficient matrices of

A

(



)

+

B

(



)

is symmetric matrices



A()+ B() is polynomialsymmetric matrix.

2) Let us assume that

A

(



)

B

(



)

is polynomial symmetric matrix. To prove that

) ( ) ( B

A = B()A().

] ... [

... ] [

) .... )(

... (

) ( ) (

0 1 1 0

0 1 1 0 0 0

1 0 1

0

B A B A B A

B B

B A A A B A

n n n n

n n n

n

  

   

   

  



 

A₀B₀ [A₀B₀]T B₀TA₀T B₀A₀ ]

[A₀B₁A₁B₀ =_[_A_B _A_B_]T 0 1 1

0 

1 0 0 1

0 1 1

0 ] [ ]

[

A B A B

B A A B

B A B A

T T T T

T T

 

.

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. n n n T n T T T n T T n T n T n T n T n n n n n n A B A B A B A B A B A B B A B A B A B A B A B A B A B A B A 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 ... ... ... ] ...[ ] [ ] [ ] ... [ ] ... [                      

Hence we get,

) ( ) ( ] ... [ ... ] [ ] ... [ . ... ] [ ) ( ) ( 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0         A B A B A B A B A B A B B A B A B A B A B A B A n n n n n n               

Conversely, assume that

A()B()B()A() and to prove )

( ) ( B

A is a polynomial symmetric matrix. Given ) ( ) ( ) ( )

( B B A

A  , we get

0 0 0

0B BA

A  , A0B1A1B0B1A0B0A1,…….

n n n n n n A B A B A B B A B A B A 0 1 1 0 0 1 1 0 ... ... ...          .

Now we use the result “XY is symmetric if and only if XY = YX”. We get A0,A1,A2,....A_n and B0,B1,B2...Bn

are symmetric matrices. Therefore all the coefficient matrices in A()B() is symmetric.

Hence A()B()B()A() is a polynomial symmetric matrix.

3) To prove A()B()+B()A()is symmetric. That is to prove . )] ( ) ( ) ( ) ( [ ) ( ) ( ) ( ) ( T A B B A A B B

A















) 1 . 3 ( ]] ... . .... ... ... [ .... ] [ ] [[ ]] ... [ ... ] [ [ ] ... [ ... ] [ [ ) ( ) ( ) ( ) ( 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 A B A B A B B A B A B A A B A B B A B A A B B A A B A B A B A B A B A B B A B A B A B A B A B A A B B A n n n n n n n n n n n n n n n                                               T n n n n n n n T T A B A B A B B A B A B A A B A B B A B A A B B A A B A B A B B A B A B A A B B A ]] ... ... [ ... ] [ ] [[ ...]] ] [ [ ...] ] [ [[ )] ( ) ( ) ( ) ( [ 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0                                   ) 2 . 3 ( ] ... ... ... .... [ ... ] [ ] [ ] ... ... [ ... ] [ ] [ ] ... ... [ ... ] [ ] [ 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 A B B A B A A B A B A B B A B A A B A B B A A B A B B A B A A B A B A B B A B A A B A B B A A B A B A B A B B A B A B A A B A B B A B A A B B A n n n n n n n T T n T T n T T n T n T T T n T T n n T T T T T T T T T T T T T n n n n n n n T T                                                         

From equation (3.1) and (3.2), . )] ( ) ( ) ( ) ( [ ) ( ) ( ) ( )

( _B _B _A _A _B _B _A T

A           Hence

) ( ) ( B 

A ₊B()A()_{is symmetric.} 4) Given n n

A

(



)











2



...





2

1

0 and are

polynomial symmetric matrices, where the coefficient matrices A₀,A₁,A₂,....A_n

are symmetric matrices. That is

T n n T T A A A A A

A0  0 , 1 1 ,..., 

) ( ] ... [ ... ... ]] ... [ [ )] ( [ 2 2 1 0 2 2 1 0 2 2 1 0 2 2 1 0               KA A A A A K KA KA KA KA KA KA KA KA A A A A K KA n n n n n T n T T T T n n T                     

E. Example 3.3

Let                 1 1 1 1 ) ( 2

A and _

         3 1 1 ) ( 2

B be polynomial symmetric matrices. Then                       4 1 2 2 2 1 ) ( ) ( )] ( ) ( [ 2 B A B A T .

F. Example 3.4

Let        2 1 ) (   

A and _

      2 1 ) (   

B be polynomial

symmetric matrices. A()B()is symmetric



T B A( ) ( )]

[



=A()B() ₍ ₎ ₍ ₎

4 3 3 1 2 2       A B           .

G. Example 3.5

Let        2 ) ( 2     A and        2 ) ( 2    

B be polynomial

symmetric matrices.

T

A B B

A( ) ( ) ( ) ( )]

[      =A()B()+ B()A()

           ) 4 ( 2 ) 2 ( 2 ) 2 ( 2 ) ( 2 2 3 3 2 4        .

Hence A()B() + B()A() is symmetric_. H. Definition 3.6

A square polynomial matrix

A

(



)

is said to be skew symmetric if

A

(



)

= __A₍₎T_{in other words all the}

A

(



)

are skew symmetric.

I. Theorem 3.7

Let A() be any square polynomial matrix. Then

T

A

A() () is polynomial skew symmetric matrix.

J. Proof ] ) ( ) ( [ ) ( ) ( ] ) ( [ ) ( ] ) ( ) ( [ T T T T T T T A A A A A A A A                

Hence T

A

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K. Example 3.8

Let

2 2 1 0

2 2

1 1

0 1 2 0

1 0 1 1

2 1

1 2 ` 1

2 1 ) (

 

    

  

A A A A

  

                    

   

 

  

  

       

     

1 2

1 1 1

1 2 1

0 0

T A and

A , 

      

     

2 1

0 0 2

0 1 0

1 1

T A and A

       

     

1 0

1 1 1

1 0 1

2 2

T

A and

A _.

             

 

           

0 1

1 0 0

1 1 0

] [

0 1

1 0

0 0 0

0

T T T

A A A

A

.

Therefore T

A

A0 0 is skew symmetric. Similarly

T A A1 1 and

T

A

A2  2 are skew symmetric. Hence T

A

A() () is polynomial skew symmetric.

L. Theorem 3.9

Let A() andB()be polynomial skew symmetric matrices

of order n. Then

1) A₍₎B₍₎is a polynomial skew symmetric.

2) KA() is a polynomial skew symmetric.

3) n

A()2 is a polynomial symmetric matrix and 1

2

) ( n

A is a polynomial skew symmetric matrix, where n is any positive integer.

M. Proof

Given

n n A A

A A

A(



) 







2 ...



2

1

0 and

n n

B B

B()   2... 

2 1

0 are polynomial skew

symmetric matrices, where the coefficient matrices

n

A A A

A₀, ₁, ₂,.... and B0,B1,B2...B_n are skew symmetric

matrices. That is

T n n T

T

A A A

A A

A₀  ₀ , ₁ ₁ ,...,  _and

. ,

,...

, 1 1

0 0

T n n T

T

B B B

B B

B   

1) An Bn n

B A B A B A

  



) ( ...

... ) ( ) ( ) ( )

( ₀ ₀ ₁ ₁

 

     

.

2) We know that “A and B are skew symmetric then A + B is skew symmetric”. From the result we have

) ( ) ( ...,

,... ) ( ) ( ), ( )

( 0 0 0 0 1 1 1 1

n n T n n

T T

B A B

A

B A B

A B A B

A

   

    

 

 Hence the

A

(



)

+

B

(



)

is skew symmetric matrices



A

(



)

+

B

(



)

is polynomial skew symmetric matrix.

3) By definition,

KA

(



)

is a polynomial skew symmetric

4) Let m be any positive integer.

Then

m m

T T

T

T T

m

A

times m A A

A

times m A A

A

times m A A A A

) ( ) 1 (

) ( )) ( ))...( ( ))( ( (

) ( ) ( ... ) ( ) (

) ( )] ( )... ( ) ( [ ] ) ( [



 



 



   

 

 

  



     

. ,

) (

, ) ( ] ) ( [

odd is m A

even is m A

A _m

m T

m

  

Hence _A₍₎m_{is symmetric when m is even and} m A() is skew

symmetric when m is odd.

N. Theorem 3.10

Let A()be a polynomial symmetric matrix. ThenA()is

diagonalizable if and only if it is complex orthogonally diagonalizable

O. Proof

Let n

n

A A

A()   2... 

2 1

0 be a polynomial

symmetric matrix.



All the coefficients matrices are symmetric. That is

n

A A A

A₀, ₁, ₂,.... are symmetric matrices.

First let us assume that A()is diagonalizable and we prove that A() is complex orthogonally diagonalizable.

)

(



A

_{is diagonalizable}



_{All the coefficients}

matrices A0,A1,A2,....A_n are diagonalizable. Suppose

0

A

is diagonalizable



there exists a nonsingular matrix P and diagonal matrix D such that

). (

, ₀ 1

0

1 

 _ _

PDP A D P A P

T T

T T T T

T T

P P DP P P D P

PDP A A PDP

  



  

 



 

1 1

1

1 0

0 1

) ( )

(

)

( _or_PPT __I_.

This says that P is orthogonal matrix. Hence

A

₀ is complex orthogonally diagonalizable. Similarly we get all the remaining coefficient matrices

n

A A

A1, 2,.... are complex orthogonally diagonalizable. Therefore

A

(



)

is complex orthogonally diagonalizable.

Conversely we assume that

A

(



)

is complex orthogonally diagonalizable, then we get All the coefficients matrices A0,A1,A2,....A_n are complex orthogonally diagonalizable. Now we show that these are diagonalizable.

Suppose

A

₀is complex orthogonally diagonalizable



there exists a orthogonal matrix P and diagonal matrix D such that 0 ,( 0 ).

T T

PDP A D P A

P   P is a orthogonal matrix

T

P

P 

 1 _{, therefore}

D P A P P A

PT   0  1

0 .

From this, we get

A

₀is diagonalizable. Similarly we get all the remaining coefficient matrices

n

A A A₁, ₂,.... are diagonalizable. Therefore A(



) is diagonalizable. Hence we proved.

P. Theorem 3.11

Let

A

(



)

be a polynomial matrix. Then

 Show that ( ) ( ) ( ) ( )

_____

  

 A A A

A if and only if A

is symmetric.

 Show that __ 

) ( ) ( ) ( ) (

_____

  

 A A A

A if and only if A

is skew symmetric.

Q. Proof

Let n

n

A A

A()   2... 

2 1

0 be a polynomial

matrix.

1) Proof of (a)

Let us first assume that ( ) ( ) ( ) ( )

_____

  

 A A A

(5)

That is















 



   

    

   



 _ _ _ _

    

) ( ) (

... ...

... ... )

( ) (

2 2 1 0

__ 2

__ 2 __

1 __

0

2 2 1 0 _____

 



 



 



A A

A A A A

n n

n n n

n A A A A

A A

A A A A A A

  

) ... (

....

... ) (

) (

__ 0 __

1 1 __ 0

2 2 __

0 __

1 1 __

2 0

1 __

0 __

1 0 0 __ 0

   

 



   



n n

A A

A A A A A

A A A

A A A A A A A A

 



) ....

... (

.... )

( ) (

0

1 1 0 2

0 2 1 1

2 0 1 0 1 0 0 0



   



 



    

  

 





 0 0 0

__

0A AA

A , ₁ ₀ ₁ ₀ ₁

__ 0 __

1

0A A A AA A A

A    

,………..

n n

n

n n

n

A A A

 

 



   

  

0 1 1 0

__ 0 __

1 1 __

0

...

... .

 

 _ _



 A A A A An An

__

1 1 __

0 0 __

... ,... ,

n

A A A₀, ₁,...

 are symmetric.

Hence

A

(



)

is polynomial symmetric matrix.

Conversely, assume that A() is polynomial symmetric matrix. This implies all the coefficient matrices

n

A A A

A0, 1, 2,.... are symmetric matrices. We know the result „X is symmetric if and only if _ 

XX X

X ‟. From this we get, 

 



 

 

 



n n

n n n n A A A

A A A

A A A A A

A A A A A A A

.. ,... ,

1 1 0 0

1 1 1 1 0 0 0

0 .















 



   

    

   



 _ _ _ _

    

) ( ) (

... ...

... ... )

( ) (

2 2 1 0

__ 2

__ 2 __

1 __

0

2 2 1 0 _____

 



 

  

 



A A

A A A A

n n n n

n n n

2) Proof of (b)

Let us first assume that ( ) ( ) ( ) ( )

_____

  

 A A A

A .

That is















 

 

    

    

   



 _ _ _ _

    

) ( ) (

... ...

... ... )

( ) (

2 2 1 0

__ 2 __

2 __

1 __

0

2 2 1 0 _____

 



 

  

 



A A

A A A A

n n n n

n n n

n A A A A

A A

A A A A A A A A A A A A



 

) ... (

....

) (

__ 0 __

1 1 __ 0

2 2 __

0 __

1 1 __

2 0 1 __

0 __

1 0 0 __ 0

    

   

 



n n n

n AA A A

A A

A A A A A A

 



) ...

( ...

.... ) (

) (

0 1 1 0

2 0 2 1 1 2 0

1 0 1 0 0 0

 

 

  

   

 



 

  



 

 0 0 0

__

0A A A

A ,

__ __

 ,………

). ...

(

...

0 1 1 0

__ 0 __

1 1 __

0

n n

n

n n

n

A A A

 

 



    

  

 

 __ __

 

A A A A An An

__ 1

1 __ 0 0 __

... ,... ,

n

A A A₀, ₁,...

 are skew symmetric.

Hence

A

(



)

is polynomial skew symmetric matrix.

Conversely, assume that A() is polynomial skew symmetric matrix. This implies all the coefficient matrices

n

A A A

A0, 1, 2,.... are skew symmetric matrices.

We know the result „X is skew symmetric if and only if





 XX

X

X ‟. From this we get,



 



 

 

  

  



  



n n

n n n n

A A

A A A A A A A A

...

,... ,

....

,... ,

1 1 0 0

1 1 1 1 0 0 0 0

.















 



    

    

   



 _ _ _ _

    

) ( ) (

... ...

... ... )

( ) (

2 2 1 0

__ 2

__ 2 __

1 __

0

2 2 1 0 _____

 



 



 



A A

A A A A

n n n n

n n n

Hence the proof

IV. CONCLUSION

Here we have extended some properties of symmetric and skew symmetric matrices to polynomial symmetric and polynomial skew symmetric matrices. All other properties can also be extended in a similar way.

REFERENCES

[1] David W.Lewis, Matrix Theory, World Scientific Publishing Co.Pte.Ltd, 1991.

[2] Gohberg I., Lancaster P., and Rodman L., Invariant Subspaces of Matrices with Applications, Wiley, New York, 1986 and SIAM, Philadelphia, 2006.

[3] Ramesh G., Sudha P.N., “On the Determinant of a Product of Two Polynomial Matrices” IOSR Journal of Mathematics(IOSR-JM) Vol.10, Issue 6, Ver.II (Nov-Dec.2014), PP 10-13.

[4] Ramesh G.1, Sudha P.N.2, Gajalakshmi R. 3”, On Normal and Unitary Polynomial Matrices” IJMER ISSN: 2249-6645,Vol.4, Iss.2, Feb.2015, 4.

[5] Roger A.Horn., Charles R.Johnson, Matrix Analysis 2nd Edition, Cambridge University Press , 1985,2013 [6] TadeuszKaczorek, Prof.drhab.in˙z., Communications

and Control Engineering, Springer- VerlagLondonLimited 2007

[7] Vardulakis A.I.G., Linear Multivariable Control, JohnWiley, Chichester, UK, 1991.