On Orthogonal Polynomial Matrices

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. IV (Mar. - Apr. 2017), PP 33-37 www.iosrjournals.org

On Orthogonal Polynomial Matrices G. Ramesh1, R. Gajalakshmi2 1

(Department of Mathemtics, Govt Arts College (Auto), Kumbakonam, Tamil Nadu, India) (Department of Mathemtics, Periyar Maniammai University, Thanjavur, Tamil Nadu, India)

2

Abstract: The concept of orthogonal polynomial matrices are introduced. Some properties and characterization for polynomial orthogonal matrices are obtained.

Keywords: Polynomial matrices, orthogonal matrices, symmetric matrices.

I.

Introduction

In matrix theory, we come across some special types of matrices and two among them are symmetric matrix and orthogonal matrix. The term orthogonal matrix was used in 1854 by Charles Hermite in the Cambridge and Dublin Mathematical Journal, although it was not until 1878 that the formal definition of an orthogonal matrix was published by Frobenius. An orthogonal matrix is the real specialization of a unitary matrix and thus always a normal matrix. Orthogonal matrices are important for a number of reasons, both theoretical and practical.

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], [4], [5].

In this paper we have introduced polynomial orthogonal matrix and extend some properties of orthogonal matrices to orthogonal polynomial matrices.

II.

Preliminaries

Definitions 2.1 [1] A matrix A is defined to be a rectangular arrangement of mn numbers arranged in m rows and n columns. If m = n tha matrix is a square matrix. AT is a matrix obtained from A by interchanging its rows into columns. If AT  A then A is said to be symmetric.A is said to be involutory if A2  I . It is orthogonal if AAT  AT A  I . It is evident that orthogonal matrices are symmetric and determinant value of an orthogonal matrix is unity.

Definition 2.2 [2] A matrix A( ) is said to be a polynomial matrix if all entries of A( ) are polynomials.

Definition 2.3 [3] T A square polynomial matrix A( ) is said to be symmetric if A( )  A( ) in other words all the coefficient matrices of A( ) are symmetric.

We state here some theorems without proof as they appeared earlier.

Theorem 2.4 [1] If A and B are two square orthogonal matrices, then (i) AB is an orthogonal matrix. (ii) A+B is not an orthogonal matrix.

Theorem 2.5 If A is orthogonal then det A   1.

III.

Polynomial Orthogonal Matrix

Definition 3. 1 A polynomial orthogonal matrix is a polynomial matrix whose coefficient matrices are orthogonal.

Example 3. 2 4 2 4 2 4  3  7 2        2   2   5 3 9 5 3 9 3 9   Let 4 2 4 3  1 2 8 A( )       2     2    2   A0  A1   A2  2 5 3  9 5 3 9 3 9  2 4 2 2 8 2  1 2     1     9 3 9 3 9  3  DOI: 10.9790/5728-1302043337

www.iosrjournals.org

33 | Page

On Orthogonal Polynomial Matrices

 3 4 0  1 2  1 1 Where A0   4  3 0  , A1   2  1 5 3   0 0 5 2 2  3 4 0  3 4 0 1 1    1 T A0 A0   4  3 0   4  3 0    0 5 5     0 0 5  0 0 5  0

2  7 4 4    1 2  , A2   4  1 8  9  1 8  1  4 0 0  1 0  I 0 1   3 4 0  3 4 0 1 0 0 1     1 T A0 A0   4  3 0   4  3 0    0 1 0   I 5 5      0 0 5  0 0 5  0 0 1

Hence

A0 A0  A0 A0  I T

T

 1 2 2   1 2 2   1 0 1    1 A A   2 1 2   2 1 2    0 1 3 3     2 2  1  2 2  1  0 0  1 2 2   1 2 2   1 0 1    1 T A1 A1   2  1 2   2  1 2    0 1 3 3     2 2  1  2 2  1  0 0 T T Hence A1 A1  A1 A1  I T 1 1

 7 4 4   7 4 4  1 1    1 A2 A2   4  1 8   4  1 8    0 9 9 8  1  4 8  1  0  4  7 4 4   7 4 4  1 1    1 T A2 A2   4  1 8   4  1 8    0 9 9 8  1  4 8  1  0  4 T T Hence A2 A2  A2 A2  I T

0  0  I 1  0  0  I 1 

0 0  1 0  I 0 1  0 0  1 0  I 0 1 

Theorem 3. 3 A polynomial orthogonal matrix is always symmetric.

Proof 2 n Let A( )  A0  A1  A2   ...........  An  be a polynomial orthogonal matrix. Here the coefficient matrices Ai ' s are orthogonal matrices.

Since orthogonal matrices are symmetric, the coefficient matrices of A( ) are all symmetric.

Hence A( ) is a symmetric matrix.

Example 3.4 Consider the orthogonal polynomial matrix,

  1  2  A( )      0 

  1  2 0

   2 1      1 0 0   0 1    Where A0   0  1 0  , A1    1 0  0 0  1 0 0    Here A0 , A1 , A2 are orthogonal.

DOI: 10.9790/5728-1302043337

0 0

A0  A1  A2 2 .

1 0 0 0    0  , A2   0 1 0  0 0 1 1    www.iosrjournals.org

34 | Page

On Orthogonal Polynomial Matrices

A0  A0 , A1  A1 , A2  A2 .

That is, A0 , A1 , A2 all are symmetric.

Hence A( ) is symmetric.

T

Also

T

Theorem 3. 5 If A( ) is a Proof

T

n n polynomial orthogonal matrix, then all its coefficient matrices are involutory.

Let A( )  A0  A1  A2   ...........  An  be polynomial orthogonal matrix. Here coefficient 2

n

matrix Ai ' s are orthogonal matrices.

That is, Since orthogonal matrices are symmetric That is,

Ai  Ai T

for i  0,1, 2, , n.

Ai Ai  Ai Ai  Ai2  I Hence each Ai is involutory.

for i  0,1, 2, , n.

T

Hence

Example 3.6

2 6   3  1       7 7  7  2 6 3  Let A( )    1         7 7 7  6 3 2    1       7 7   7  0 1 0   3 2   1 Where A0    1 0 0  , A1    2 6 7  0 0  1   6 3  0  1 0  0  1 0   1 0     2 A0    1 0 0   1 0 0    0 1  0 0  1 0 0  1  0 0    

A0  A1

 6   3  2 

0  0  I 1 

A0  I 2

Hence

 3  2  6  3  2  6 1 0 0 1     1 A1    2 6  3    2 6  3    0 1 0   I 7 7       6  3  2   6  3  2  0 0 1 2 Hence A1  I 2

Hence all coefficient matrices A0 ,

A1 are involutory.

Theorem 3. 7

If A( ) is a polynomial orthogonal matrix if and only if  A( )  is polynomial orthogonal matrix.

T

Proof Let A( )  A0  A1  A2   ...........  An  be polynomial orthogonal matrix. Here coefficient 2

n

matrix Ai ' s are orthogonal matrices. That is,

 A0 A0  I   T T A1 A1  A1 A1  I  . .......... .......... .......... T T An An  An An  I   A0 A0

T

T

DOI: 10.9790/5728-1302043337

3.1

www.iosrjournals.org

35 | Page

On Orthogonal Polynomial Matrices To prove  A( ) T is polynomial orthogonal matrix.

From 3. 1

A()T  A0T  A1T   A2T 2 ................. AnT n

We know that coefficient matrices

,

Ai s are orthogonal matrices.

Hence  A( )  is a polynomial orthogonal matrix.

Similarly we can prove the converse.

Example 3.8 Consider the orthogonal polynomial matrix, T

4 97   3 30      5 102   A  A  A( )   5 97 0 1  4  97   3  30     5 97   5 102 4 97   3 30      T 5 102   A  A  A( )   45 97 0 1   97   3  30     5 97   5 102 T Here A    A( )  30 3 4   97 1 Now A( )    5  4  3   97   102

97   102  30   97  97   30   1 3 4  Where A0    , A1   97 102  5  4  3  97  30    97   102 1  3 4  1  3 4  1 0 T    I .

A0 A0   5  4  3  5  4  3   0 1  Also

A0  A0 .

T

97  30  30   A A   97 102  97  97  30  97   97  102  102 T Also A1  A1 .

T 1 1

97   102    1 0   I   30    0 1  97 

Theorem 3. 9 If A( ) is a polynomial orthogonal matrix then determinant of Proof:

A( ) is a polynomial in  .

Let A( )  A0  A1  A2   ...........  An  be polynomial orthogonal matrix. Here coefficient 2

n

matrix Ai ' s are orthogonal matrices. From theorem 2.5, we have det Ai   1,  i  0, 1, , n . Then all the coefficient matrices have determinant value  1 .

Hence det  A( )  

i  det Ai   n

i0

  1    2   n .

It is a polynomial in  .

DOI: 10.9790/5728-1302043337

www.iosrjournals.org

36 | Page

On Orthogonal Polynomial Matrices Example 3.10

 1   2  (i). Let A( )   1  2 

 1  1  2 1     1  2  2 1

det A( ) 

det A( )  det A( )  det A( ) 

det A( ) 

1   2   A  A  If 0 1 1     2 

(1  2 )(1  2 )  1  1  2  2  2  1 2

1

2 1 1 2 2

2

2

2



2

  1  2  1 2

2

2  1 0 0   A0 0 (ii ) Let A( )   0 1   det A( )  1  0

(i.e) constant Hence determinant of a polynomial orthogonal matrix is a polynomial.

Remark 3. 11 The product of two polynomial orthogonal matrices need not be orthogonal.

For, Let A( ) and B( ) be two polynomial orthogonal matrices.

Let

A( )  A0  A1  A2 2  ...........  An n

polynomial orthogonal matrices.

Where A0 , A1 , A2 , ..........., To prove A( ) B( ) orthogonal.

and

B( )  B0  B1  B2 2  ...........  Bn n

be

An and B0 , B1 , B 2 , .........., B n are orthogonal matrices.

A( ) B( )  ( A0  A1  A2 2  ...........  An n )(B0  B1  B2 2  ...........  Bn n )  A0 B0  ( A1 B0  B1 A0 )  .......... ( A0 Bn  A2 Bn 1  ........ An B0 )n By the theorem (2.4) product of two orthogonal matrices are orthogonal.

So A0 B0 is orthogonal matrix. But the second term of the above equation is not possible.Because sum of the two orthogonal matrices need not be orthogonal. Hence the product of two polynomial orthogonal matrices need not be orthogonal

IV.

Conclusion

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

References [1].

[2].

[3].

[4].

[5].

David W.Lewis, Matrix Theory (World Scientific Publishing Co.Pte.Ltd, 1991).

I. Gohberg,, P. Lancaster and L. Rodman, Invariant Subspaces of Matrices ith Applications (Wiley, New York, 1986 and SIAM, Philadelphia, 2006).

G. Ramesh, P.N. Sudha, “On Polynomial Symmetric and Polynomial Skew Symmetric Matrices” IJSRD - International Journal for Scientific Research & Development| Vol. 3, issue 06, 2015 | ISSN (online): 2321-0613 A.I.G. Vardulakis, Linear Multivariable Control (JohnWiley, Chichester, UK, 1991).

W.A. Wolovich, Linear Multivariable Systems (Springer Verlag, 1974).

DOI: 10.9790/5728-1302043337

www.iosrjournals.org

37 | Page