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ON K-CIRCULANT MATRICES INVOLVING THE FIBONACCI
Also in [17]
M. Abreu
a , D. Labbate b , R. Salvi c , N. Zagaglia Salvic,? aDipartimento di Matematica, Università della Basilicata, C. da Macchia Romana, 85100 Potenza, Italy
b Dipartimento di Matematica, Politecnico di Bari, I-70125 Bari, Italy cDipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, I-20133 Milano, Italy
Received 31 July 2006; accepted 26 February 2008
Available online 11 April 2008
Submitted by R.A. Brualdi
Abstract
in terms of circulant and retrocirculant block(0,1)-matrices in which each block contains exactly one or
two entries 1. In particular, we prove that a generalizedk-circulant matrixAof composite ordern=kmis symmetric if and only if eitherk=m-1ork≡0ork≡1 modm, and we obtain three basic symmetric© 2008 Elsevier Inc. All rights reserved.Keywords:Generalized circulant matrix; Block circulant matrix; Centrosymmetric matrix; Persymmetric matrix
1. Introduction
A matrix of ordernis said to beh-circulant,1?hCorresponding author.
E-mail addresses:abreu@unibas.it(M. Abreu),labbate@poliba.it(D. Labbate),rodolfo.salvi@polimi.it(R. Salvi),
norma.zagaglia@polimi.it(N. Zagaglia Salvi).0024-3795/$ - see front matter
(2008 Elsevier Inc. All rights reserved.doi:10.1016/j.laa.2008.02.033brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Elsevier - Publisher Connector
368M. Abreu et al. / Linear Algebra and its Applications 429 (2008) 367-375
one by shifting every element 1 position to the left. LetP n be the circulant matrix having as first row(010...0). If there is no possibility of ambiguity we often drop the subscriptnand simply writeP n asP. In [3] a matrix A of ordern=kmand(n,h)=kis calledgeneralizedh-circulantwhen it is partitioned intoh-circulant submatrices of typem×n. LetA j ,1?j?k,betheh-circulant submatrix of A of typem×n, formed by the rows1+(j-1)m,2+(j-1)m,...,jm;
thenAis partitioned inkblocks of typem×nas follows:? ??A 1 A 2 A k In the same paper the following characterization of generalizedh-circulant matrices is shown. Theorem 1.1.A matrix A of ordernis generalizedh-circulant,where(n,h)=kandn=km, if and only if it satisfies the relation AP h =P A, whereP is direct sum ofkmatrices coinciding withP m ,i.e.P =diag(P m ,...,P m A matrixAof ordern=km, partitioned intom×nsubmatricesA j ,1?j?k, is said to be(m×n)-block circulantwhen every block, different from the first one, is obtained from the preceding one by shifting every column one position to the right. This condition is equivalent to say thatA j+1 =A jP, where 1?j?k-1.
The following is an example of a generalized 2-circulant matrix and a(2×4)-block circulant matrix both of order 4? ??1234 34125678
7856?
??and? ??1234 5678
4123
8567?
A matrixAof ordern, where(n,h)=kandn=km, is calledblock generalizedh-circulant if it is generalizedh-circulant and(m×n)-block circulant. An example of a block generalized 2-circulant matrix is? ??1234 3412
4123
2341?
A(0,1)-matrixA=[a
i,j ]is a permutation matrix if there exists a permutationσsuch that a i,j =1 if and only ifj=σ(i). If necessary, we useA(σ)to emphasizeσ. We will represent a permutation by the one-line notationσ=(σ(1)σ(2)...σ(n))[1]. Permutation matrices partitioned into particular submatrices are considered in [5]. Note that the circulant matrixP:=P n is always the permutation matrix asssociated to the permutation σ=(1...n). SincePis a permutation matrix, thenP T =P -1 ,P n =I n and(P T s-1 =P s whenn=2s-1. M. Abreu et al. / Linear Algebra and its Applications 429 (2008) 367-3753693- or 5-circulant permutation matrices. Hence, it seems worthwhile to characterize the order and
the structure of symmetric generalizedk-circulant matrices of ordern=km. We present such a characterization in Section2. Furthermore, we extend such a characterization tocentrosymmetric matrices in Section3.2. Symmetric matrices
LetAbe a generalizedk-circulant permutation matrix of composite ordern=km. LetA b [B i,j ]be the matrix obtained by partitioningAinto submatrices (blocks)B i,j of orderm. Recall that the matrixE i,j of orderndenotes the matrix having all the elements zero, buta1inposition (i,j). We will also consider thedistancebetween theith andjth rows (or columns) of a matrix A to be|i-j|.Consecutiverows (or columns) are those at distance 1. Also the first and last row (or column) of a matrix of ordermare consideredconsecutive modulo m.TheithrowandithcolumnofA
b aredenotedbyR i =[B i,1 ,B i,2 ...B i,k ]andC i =[B 1,i ,B 2,i ...,B k,i T , with 1?i?k, respectively. The following is an immediate consequence of these definitions. Lemma 2.1.LetAbe a generalizedk-circulant permutation matrix.Then two ones ofR i belong i belong to rows having distance a multiple ofk.We denote byP
b :=[P b i,j ]the block permutation(0,1)-matrix of orderkwith blocks of order msuch that P b i,j :=?I m ifj≡i+1 modk O m otherwiseNote:The matrixP
b has the same structure as the permutation matrixP k in which every entry1 is substituted by the identity matrixI
m of orderm, and every entry 0 is substituted by the null matrixO m of orderm.Theorem 2.2.LetA=[a
i,j n=km,with1···E
m-1,2 E m-1,2 E 1,m +E m,1···E
i-1,m-i···E
m-2,3 E 2,m-1 E 3,m-2···E
m-1,2···E
1,m +E m,1Ifmis odd,saym=2s-1,wheres>1,A
b coincides either withK 1 or with the circulant block matrix K 2 ??E s,s E s+1,s+m-1···E
1,m +E m,1···E
s-1,s-m+1 E s-1,s-m+1 E s,s···E
m-1,2···E
s-2,s-m E s+1,s+m-1 E s+2,s-2···E
2,m-1···E
s,sMoreover,K
2 :=K 1 P sb370M. Abreu et al. / Linear Algebra and its Applications 429 (2008) 367-375
Proof.Note that thek-circulant submatrixR
1 =[B 1,1 ,B 1,2 ,...,B 1,k ], containsmones. Since k[PDF] generate categorical variable stata
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