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Block Circulant Matrix with Circulant Polynomial Matrices As Its Blocks G.Ramesh, * R.Muthamilselvam,** * Associate Professor of Mathematics, Govt

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Block Circulant Matrix with Circulant Polynomial Matrices As Its Blocks G.Ramesh, * R.Muthamilselvam,** * Associate Professor of Mathematics, Govt 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 a12, a ,..., an be an ordered n-tuple of polynomial with complex coefficients , and let them generate the circulant polynomial matrix of order n 5 : a12 a ... an a a ... a A n 12 1.1 a2 a 3 ... a 1 We shall often denote this circulant polynomial matrix as A circ a , a ,..., a 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 Volume IX Issue IV APRIL 2020 Page No : 519 Science, Technology and Development ISSN : 0950-0707 The symbol * designates the conjugate transpose. From the spectral mapping theorem, we may represent A in the form A a a a 21 ... a n 1.7 1 2 3 n Where is the permutation matrix circ 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 of order mn is meant an mn mn polynomial matrix of the form AAA12 ... m AAA ... bcirc A, A ,..., A mm11 12 m AAA2 3 ... 1 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 A1, A 2 ,..., An m A k 1 1.10 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 . Volume IX Issue IV APRIL 2020 Page No : 520 Science, Technology and Development ISSN : 0950-0707 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 A circulant polynomial matrices as its blocks analogous to that of the results found in 1,4,6,9,10 . Definition: 2.1 Let 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 CPBCm,n . Example: 2.2 3 12 1 1 7 10 3 1 12 10 1 7 1 7 10 3 12 1 A 10 1 7 3 1 12 12 1 1 7 10 3 1 12 10 1 7 3 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 Volume IX Issue IV APRIL 2020 Page No : 521 Science, Technology and Development ISSN : 0950-0707 Proof: i Fm F n m I n FFFFFIF ** m n m m m n n n ** FFFFIFFm n m n m n m n * FFFFIFFm n m n m n m n m IFF n m n * ()iim I n F m F n FFFIFFF**** m m m n n n m n ** * FFIFFFFm n m n m n m n * FFI m n m n Lemma: 2.5 * If AFFFF m n m n where is diagonal polynomial matrix, then A CPBCm,n or equivalently, that A commutes with both mn I and Imn . Proof: Now AI mn * FFFFIm n m n m n * FFIFFm n m n m n * FFIFFm n m n m n * m IFFFF n m n m n . mn IA Volume IX Issue IV APRIL 2020 Page No : 522 Science, Technology and Development ISSN : 0950-0707 Also, AImn * FFFFIm n m n m n * FFIFFm n m n m n * FFIFFm n m n m n * IFFFFm n m n m n IA . mn Theorem: 2.6 All polynomial matrices in CPBCm,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 ,km 0,1,..., 1, 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 m n m n where is diagonal polynomial matrix is in CPBCm,n . Proof: Assume that A is a block circulant polynomial matrix. m1 k From 1.10 , 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 Volume IX Issue IV APRIL 2020 Page No : 523 Science, Technology and Development ISSN : 0950-0707 2i polynomial matrix of order m, diag 1, , km ,..., 1 where e m m1 **k Hence, AFFFF m m m n k1 n k 0 m1 **k FFFFm n m k1 m n k 0 m1 * k FFFFm n m k1 m n k 0 * AFFFF m n m n Conversely, assume that where is diagonal polynomial matrix. We have to prove that A BCCBm,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 m n m n 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 AIIBABm n m n m n . Proof: AIAIAAIIm n m n m m n n 22 AImn 2 2 AIAIm n m n k k By induction, AIAIm n m n Volume IX Issue IV APRIL 2020 Page No : 524 Science, Technology and Development ISSN : 0950-0707 j j Similarly, IBIBm n m n kj kj Now AIIBAIIBm n m n m n m n kj AIIBm m n n 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 1.10 k Therefore, by , we have AA mk 1 k 0 Where the blocks Ak1 are themselves circulant polynomial matrices. Then n1 j Aak1 k 1, j i n j0 mn11 kj Now Aa m k1, j 1 n kj00 mn11 kj m a k1, j 1 n kj00 mn1, 1 kj ak1, j 1 m n 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. Volume IX Issue IV APRIL 2020 Page No : 525 Science, Technology and Development ISSN : 0950-0707 (ii) A circulant polynomial matrix of level 2 is in . (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 1.10 , 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 Aak1, j 1 k 1, j 1, r 1 p 2.3 r0 Combining 2.1 ,2.2 and 2.3 we have mn11p1 k j r Aa m n k1, j 1, r 1 p 2.4 k0 j 0 r 0 mn11p1 k a j r CPBC m k1, j 1, r 1 n p m,n k0 j 0 r 0 m1, n 1, p 1 k j r ak1, j 1, r 1 m n p k, j , r 0 Since, k*** k j k r k mmmmmnnn F F , F F and pppp F F 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 Volume IX Issue IV APRIL 2020 Page No : 526 Science, Technology and Development ISSN : 0950-0707 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.
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