Science, Technology and Development ISSN : 0950-0707
Block Circulant Matrix with Circulant Polynomial Matrices as its Blocks G.Ramesh, * R.Muthamilselvam,** * Associate Professor of Mathematics, Govt. Arts College(Autonomous), Kumbakonam. ([email protected]) ** Assistant Professor of Mathematics, Arasu Engineering College, Kumbakonam. ([email protected]) ______Abstract: The characterization of block circulant matrix 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. 2i 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 nn12 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 permutation matrix 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 circ0,1,0,...,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
m1 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 AImn
* FFFFImnmnmn
* FFIFFmnmnmn
* FFIFFmnmnmn
* m IFFFF n m n m n
. mn IA
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Also, AImn
* 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
k1 ,0,1,...,1,km then the diagonal polynomial matrix of the eigen values of the
m1 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.
m1 k From 1 . 1 0 , can be written as AA mk 1 where the blocks k 0
are AAA12,,...,. m The Ak1 are circulant polynomial matrices if and only if
* AFFk11 n k n , where Fn is the Fourier polynomial matrix of order n
and k1 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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2i polynomial matrix of order m, diag1,,,..., km 1 where e m
m1 **k Hence, AFFFF m m m n k1 n k 0
m1 **k FFFFmnmkmn 1 k 0
m1 * 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 ACPBCm,n . Then A is a polynomial (of two variables) in
mn I and mn .
Proof:
Since A is a block circulant polynomial matrix.
m1 1 . 1 0 k Therefore, by , we have AA mk 1 k 0
Where the blocks Ak1 are themselves circulant polynomial matrices. Then
n1 j Aakkj11, in j0
mn11 kj Now Aa mkjn 1, 1 kj00
mn11 kj mkjn a 1,1 kj00
mn1,1 kj akjmn1,1 kj,0
mn1, 1 kj ak1, 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 Polynomials 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). m1 k From 1 . 1 0 , we have AA mk 1 2.1 k 0
Where each Ak1 is a level 2 circulant polynomial matrix of type (n,p). n1 j Thus, AAk1 n k 1, j 1 2 . 2 j0
Where each Akj1,1 is a circulant polynomial matrix of level 1 and of order p. Thus, from 1.7 p1 r Aakjkjrp1,11,1,1 2.3 r0 Combining 2.1 ,2 . 2 and 2.3 we have mn11p1 kjr Aa mnk j rp 1, 1, 1 2 . 4 kjr000 mn11p1 kjr mkj rnp a 1, 1, 1 kjr000
mnp1, 1, 1 kjr akjrmnp1, 1, 1 k, j , r 0 Since, kkjkrk *** mmmmmnnnpppp FFFFandFF ,
There fore, m1, n 1, p 1 ***k k k A akjr1, 1, 1 F mmm F F nnn F F ppp F k, j , r 0
m1, n 1, p 1 a F** F F k k k k1, j 1, r 1 m n p m n p k, j , r 0 ** FFFm n p
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m1, n 1, p 1 * a F F F k k k k1, j 1, r 1 m n p m n p k, j , r 0 FFFm n p
m1, n 1, p 1 * a k k k FFFm n p k1, 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
bylemma2.11 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 mnp1, 1,1 kjr AaIIII kjrmnpmnpm1, 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 mn11p1 kjr Aa mnk j rp 1, 1, 1 kjr000
mn11p1 kjr mkj rnp a 1, 1, 1 kjr000 mnp1, 1, 1 kjr a kjrmnp1, 1, 1 k, j , r 0 Hence, by 2.14 & lemma2 . 1 3, the above can be written as
m1, n 1, p 1 k j r A akjr1, 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.
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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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