Science, Technology and Development ISSN : 0950-0707

Block Circulant with Circulant Matrices as its Blocks G.Ramesh, * R.Muthamilselvam,** * Associate Professor of , Govt. Arts College(Autonomous), Kumbakonam. ([email protected]) ** Assistant Professor of Mathematics, Arasu Engineering College, Kumbakonam. ([email protected]) ______Abstract: The characterization of block with circulant polynomial matrices as its blocks are derived as a generalization of the block circulant matrices with circulant block matrices. Keywords: Circulant polynomial matrix, Block circulant polynomial matrix, Circulant block polynomial matrix. AMS Classification: 15A09, 15A15, 15A57.

I. Introduction

Let aaa12,,...,  n   be an ordered n-tuple of polynomial with complex coefficients , and let them generate the circulant polynomial matrix of order n 5 :

aaa12   ... n    aaa   ...   A   n 12 1.1      aaa231   ...   We shall often denote this circulant polynomial matrix as Acirc aaa ,,..., 1.2    12    n     It is well known that all circulant polynomial matrices of order n are simultaneously diagonalizable by the polynomial matrix F  associated with the finite Fourier transforms. 2i Specifically, let   exp   ,i   1 1 . 3 n 1 1 1 ... 1 21n 1       1       and set Fn    2 1 2   4   2(n 1)    1.4   nn12 1    ...    The Fourier polynomial matrix F  depends only on n. This matrix is also symmetric polynomial and unitary polynomial FFFFI         and we

have AFF        1.5

Where  diag  12,  ,..., n  1.6

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The symbol * designates the conjugate transpose. From the spectral mapping theorem, we may represent A in the form

Aaaaa 21... n 1.7   123          n       Where  is the c i r c 0 ,1 ,0 ,0 ,. . .    . Also, let A be an nn polynomial matrix. Then is a circulant polynomial

matrix if and only if AA        1 . 8

The matrix    circ0,1,0,...,0

From the diagonalization of the circulant polynomial matrix , we have

FF 1.9           Let AAA,,..., be a collection of square polynomial matrices,  12    n   each of order n. By a block circulant polynomial matrix 2 ,3 ,7 ,8  of type mn,  and oforder mn  is meant an mnmn polynomial matrix of the form

AAA12   ... m    AAA   ...   bcirc AAA,,...,  mm11  12    m     AAA231   ...  

We denote the set of all block circulant polynomial matrices of order mn as BCm,n  . A representation of block circulant polynomial matrices can be developed as

m1 k bcirc AAAA121,,...,  nmk        1 . 1 0 k 0

Also, let be of type :

AAA11 12  1m    A    AAAm12 m  mm   m m block matrixeachblock isof oreder n is a circulant block polynomial matrix 7,5 if

each block Aij   is a circulant polynomial matrix.

We denote the set of all circulant block polynomial matrices of order n as CBm,n  .

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This paper is devoted to the study of block circulant matrix with circulant polynomial matrices as its blocks.

II Block Circulant Matrix with Circulant Polynomial Matrices as its Blocks

In this section, we have given a characterization of block circulant matrix with circulant polynomial matrices as its blocks analogous to that of the results found in 1,4 ,6 ,9 ,1 0 . Definition: 2.1

Let A be of type (m,n). is said to be a block circulant matrix with circulant polynomial matrices as its blocks if it is circulant block wise and each block is a

circulant polynomial matrix and is denoted by C P B Cm,n  .

Example: 2.2

31211 710    3112101 7 1 7103121  A   101 73112  1211 7103  112101  73

is in CPBCm,n  .

Remark: 2.3 A polynomial matrix in is not necessarily a circulant polynomial

matrix.

Lemma: 2.4

Fm  and Fn  satisfies the following equalities.

(i) FFIIFFm n     m   n      m   n    m   n   

** (ii) m  IFFFFI n    m   n     m   n     m   n    

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Proof:

iFFI mnmn     

FFFFFIF ** mnmmmnnn                 

 **     FFFFIFFm  n   m   n    m   n    m   n   

* FFFFIFF mnmnmnmn                

 mnmn IFF     

* ()iiIFF mnmn       

FFFIFFF****   mmmnnnmn                

** * FFIFFFFmnmnmnmn              

* FFI        mnmn   

Lemma: 2.5

* If AFFFF  mnmn          where  is diagonal

polynomial matrix, then A CPBC m,n   or equivalently, that A commutes with both

mn  I   and Imn   .

Proof:

Now AImn   

* FFFFImnmnmn             

* FFIFFmnmnmn             

* FFIFFmnmnmn             

* m  IFFFF n    m   n      m   n    

. mn  IA   

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Also, AImn   

* FFFFImnmnmn             

* FFIFFmnmnmn              

* FFIFFmnmnmn             

* IFFFFmnmnmn             

IA . mn      Theorem: 2.6

All polynomial matrices in C P B Cm,n  are simultaneously diagonalizable

polynomial matrices by the unitary polynomial matrix FFmn   , and they commute. If the eigen values of the circulant block polynomial matrices are given

k1 ,0,1,...,1,km then the diagonal polynomial matrix of the eigen values of the

m1 k CPBC  polynomial matrix is given by mk  1   . Conversely, any matrix of k 0

* the form AFFFF  mnmn          where  is diagonal

polynomial matrix is in CPBCm,n  .

Proof:

Assume that A is a block circulant polynomial matrix.

m1 k From 1 . 1 0 , can be written as AA  mk  1   where the blocks k 0

are AAA12,,...,.  m   The Ak1  are circulant polynomial matrices if and only if

* AFFk11  n  k  n  , where Fn  is the Fourier polynomial matrix of order n

and k1  is a diagonal polynomial matrix of order n.

From 1.9 , we have kk FF*    where   is the     m  m  m     m    

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2i  polynomial matrix of order m,  diag1,,,...,   km  1   where    e m

m1 **k Hence, AFFFF  m   m  m   n   k1   n    k 0

m1 **k FFFFmnmkmn      1       k 0

m1 * k FFFFmnmkmn       1       k 0

* AFFFF  mnmn         

Conversely, assume that where is diagonal polynomial matrix.

We have to prove that A BCCB m,n   . It is enough to prove that A commutes

with both mn  I   and Imn   .

From lemma 2. 5 , we have AIIA m  n    m   n     and

AIIA mnmn         

Hence, is a block circulant matrix with circulant polynomial matrices as

its blocks.

Lemma: 2.7

Let j, k be nonnegative integers. Let ABmm,   be of order m and n. Then

kjkj AIIBABmnmnmn            .

Proof:

AIAIAAII                m n m n m m n n AI22   mn

2 2  AIAImnmn      

k k By induction,  AIAIm  n   m   n   

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j j Similarly, IBIBmnmn      

kj   kj Now AIIBAIIBmnmnmnmn               

kj AIIBmmnn      

kj ABmn   .

Theorem: 2.8

Let ACPBCm,n  . Then A is a polynomial (of two variables) in

mn  I   and  mn    .

Proof:

Since A is a block circulant polynomial matrix.

m1 1 . 1 0 k Therefore, by   , we have AA  mk  1   k 0

Where the blocks Ak1  are themselves circulant polynomial matrices. Then

n1 j Aakkj11, in       j0

mn11 kj Now Aa  mkjn  1, 1     kj00

mn11 kj mkjn  a 1,1     kj00

mn1,1 kj  akjmn1,1      kj,0

mn1, 1 kj      ak1, j 1    m   I n    I m    n    kj,0 by lemma (2.7)

This is a polynomial in nn   I   and Imn   . Remark: 2.9

(i) A circulant polynomial matrix of level 1 is an ordinary circulant polynomial matrix.

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(ii) A circulant polynomial matrix of level 2 is in C P B Cm,n  . (iii) A circulant polynomial matrix of level 3 is a block circulant polynomial whose block are level 2 circulant polynomial matrices.

Theorem: 2.10

A circulant polynomial matrix of level 3 and type (m,n,p) is diagonalizable

polynomial matrix by the unitary polynomial matrix FFFm  n  p  . Proof : Let A be a level 3 circulant polynomial matrix of type (m, n, p). m1 k From 1 . 1 0 , we have AA  mk  1    2.1 k 0

Where each Ak1  is a level 2 circulant polynomial matrix of type (n,p). n1 j Thus, AAk1   n  k  1, j  1   2 . 2 j0

Where each Akj1,1  is a circulant polynomial matrix of level 1 and of order p. Thus, from 1.7 p1 r Aakjkjrp1,11,1,1       2.3 r0 Combining 2.1 ,2 . 2 and 2.3 we have mn11p1 kjr  Aa  mnk  j rp  1,  1,  1     2 . 4 kjr000  mn11p1 kjr  mkj  rnp a 1,  1,  1       kjr000 

mnp1,  1, 1 kjr  akjrmnp1,  1,  1        k, j , r 0 Since, kkjkrk *** mmmmmnnnpppp  FFFFandFF     ,                

There fore, m1, n  1, p  1 ***k k k A  akjr1,  1,  1    F mmm    F   F nnn    F   F ppp     F   k, j , r 0

m1, n  1, p  1 a F**  F   F  k    k    k    k1, j  1, r  1   m  n  p  m  n  p   k, j , r 0 ** FFFm  n  p   

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m1, n  1, p  1 * a F  F   F  k    k    k    k1, j  1, r  1   m  n  p   m  n  p   k, j , r 0 FFFm  n  p   

m1, n  1, p  1 * a  k    k    k    FFFm  n   p     k1, j  1, r  1  m n p  k, j , r 0 FFFm  n  p   

Thus, a circulant polynomial matrix of level 3 and type (m, n,p) is diagonalizable polynomial matrix.

Lemma: 2.11

ABAIIBm n   m   n    m   n    Proof:

AIIB mnmn       AIIBmmnn      

ABmn   Lemma: 2.12

ABCmnp   

 AIIBIICmnpmnPmnP             Proof:

ABCABCm n   p   m   n   p   ABIIC   mnpmnp            ABIICm  n   p   mn   p    AIIBIIC   mnpmnpmnp              

bylemma2.11 Lemma: 2.13

For nonnegative integers k, j, r k j r k j ABCAIIBIm  n   p   m   np   m   n   P  

r  ICmn  p   Proof: By the lemma 2.12 ABCAIIBIICk j   r   k      j      r  m  n  p   mn  np  m    P  mn  p  

k k  We have AIAIm  np   m   np   

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j j  IBIIBImnpmnp           r ICIC r       mnpmnp  k kjr  Hence, ABCAImnpmnp         jr IBIIC         mnpmnp

Theorem: 2.14 Let A be of type m,, n p  and be a circulant polynomial matrix of level 3. Then mnp1, 1,1 kjr    AaIIII  kjrmnpmnpm1, 1, 1, mp                  k, j , r 0

Proof: Since A is of type m,, n p  and is a circulants polynomial matrix of level 3. Therefore, from 2 . 4 it can be written as mn11p1 kjr  Aa  mnk  j rp  1,  1,  1     kjr000

mn11p1 kjr  mkj   rnp a 1,  1,  1       kjr000 mnp1, 1, 1 kjr  a kjrmnp1, 1, 1        k, j , r 0 Hence, by 2.14 & lemma2 . 1 3, the above can be written as

m1, n  1, p  1 k j r A  akjr1,  1,  1   m   I np   I m   n   I p   I mn   p   k, j , r 0

Conclusion:

Some of the characterizations of block circulant matrix with circulant polynomial matrices as its blocks are discussed here.

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References: 1 Ahlberg, J.H. Block Circulants of level K.Division of applied Mathematics, Brown University, Providence, R.I, June 1976. 2 Ahlberg, J.H. Doubly Periodic Initial value Problems, Lecture notes, 1997.

3 Chao, Chong -Yun. On a Type of Circulants, and Its Appl.,   Vol .6 (1973), PP. 241-248. 4Chao, Chong -Yun. A note on Block Circulant Matrices, Kyungpook Math .J., Vol. 14 (1974). 5 Davis,P.J. Circulant Matrices, New York, 1979.

6 Horn, R.A., and C.R.Johnson. Matrix Analysis, Cambridge University press, Newyork, 2013. 7 Ramesh. G., and Muthamilselvam.R. A Study on g-Circulant Polynomial Matrices, The Intenational Journal of analytical and experimental modal Analysis, Volume XI, Issue XII, December /2019, PP.3284-3297. 8 Ramesh. G., and Muthamilselvam.R. On Block Circulant Polynomial Matrices, Infokara Research, Volume 9, Issue 2, 2020, PP. 559-566. 9 Smith, R.L. Moore –Penrose Inverses of Block Circulant and Block k-Circulant Matrices, Linear Algebra and its Appl., Vol.16 (1997), PP. 237-245. 10 Trapp, G.E. Inverses of Circulant Matrices and Block Circulant Matrices, Kyungpook Math.J., Vol.13 (1973), PP.11-20.

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