On sign symmetric circulant matrices

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Abstract

In the last four decades many researchers have studied and analyzed the study of sign symmetry and positivity of principal minors of matrices, since these issues are related to stability. In this work we extend the theory about sign symmetric basic p-circulant permutation and sifted p-circulant matrices. We present and prove sufficient and necessary conditions for P-matrices and necessary conditions for P2-matrices. Finally we present a class of matrices, where the P2-matrices are stable.

Section snippets

Introduction and preliminaries

Consider two subsets α and β of {1,2,,n} with the same cardinality (|α|=|β|) and an n×n square real matrix A. We denote by A[α|β] the minor with the rows indexed by α and columns indexed by β. If α=β the minor is a principal minor of A. The matrix A is called sign symmetric if A[α|β]A[β|α]0, for all α and β{1,2,,n} with |α|=|β|. The matrix A is called weakly sign symmetric if A[α|β]A[β|α]0, for all α and β{1,2,,n} with |α|=|β|=|αβ|+1.

A square real matrix A is called a Q-matrix (Q0-matrix

Sign symmetry of circulant permutation matrices

Definition 2.1

A n × n real matrix is called a circulant matrix if it is of the formCn=a1a2a3anana1a2an-1an-1ana1an-2a2a3a4a1.

The following lemma for circulant matrices is well known [8].

Lemma 2.1

Let ρi be the ith of the n roots of unity. The eigenvalues of the circulant matrix (2.1) are given byλi=k=1nakρik-1,i=1(1)n1.

On shifted circulant permutation matrices

Hershkowitz and Keller [7] proved that the matrixA=x1y100000yn-1yn00xn,where the xi’s share the same sign and i=1nyi>0, in case n is even, is neither sign symmetric nor anti sign symmetric. However, this is not true in a more general case. For example, let the matrixA4,2=x10y100x20y2y30x300y40x4.Since aijaji0,i=1(1)4, and j=i+2(mod4), it is easy to verify that

  • (i)

    When α and β are subsets of {1,2,3,4} of cardinality +1A4,2[α|β]A4,2[β|α]=yi+1yi+3,i=0,1.

  • (ii)

    When α and β are subsets of {1,2,3,

On positivity of principal minors of a shift circulant matrix A2n,2

Let x,yIR. The A2n,2 shift circulant matrix has the formA2n,2=x0y0000x0y00000x0yy000x00y000x.

Theorem 4.1

Let A2n,2 be a shift circulant matrix, with x,yIR. This matrix is a P-matrix if and only if:

  • (i)

    x>0,x+y>0, if n odd.

  • (ii)

    x>0,x2-y2>0, if n even.

Proof

The graph of A2n,2 is of the form shown in Fig. 1. This means that there exists a permutation matrix P, so that the product P-1A2n,2P will have a block diagonal form, where the diagonal elements are the same with shift circulant basic matrix Cn and where det(Cn)=xn+

The shift circulant matrix A6,2

Let the shift circulant matrixA6,2=x0y0000x0y0000x0y0000x0yy000x00y000x.We denote D|α|=A6,2[α|β]A6,2[β|α], where α,β{1,2,3,4,5,6}, with |α|=|β|. We have n|α|=6|α| sets α and n|α|2 products D|α|. So, there exist |α|=16D|α|=430 products of the form A6,2[α|β]A6,2[β|α], with αβ. From these products, +66 are different from zero and are distributed as follows:

  • There are +6 products, D50, of the formD5=-xy(x+y)2(x2-xy+y2)2y2.

  • There are +36 products, D40, of the formsD4=-x3y5,(18cases),orx4y6,(18

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