On sign symmetric circulant matrices
Section snippets
Introduction and preliminaries
Consider two subsets α and β of with the same cardinality () and an square real matrix A. We denote by the minor with the rows indexed by α and columns indexed by β. If the minor is a principal minor of A. The matrix A is called sign symmetric if , for all and with . The matrix A is called weakly sign symmetric if , for all and with .
A square real matrix A is called a Q-matrix (-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 form Lemma 2.1 Let ρi be the ith of the n roots of unity. The eigenvalues of the circulant matrix (2.1) are given by
On shifted circulant permutation matrices
Hershkowitz and Keller [7] proved that the matrixwhere the xi’s share the same sign and , 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 matrixSince and , it is easy to verify that
- (i)
When α and β are subsets of of cardinality +1
- (ii)
When α and β are subsets of
On positivity of principal minors of a shift circulant matrix A2n,2
Let . The shift circulant matrix has the form Theorem 4.1 Let be a shift circulant matrix, with . This matrix is a P-matrix if and only if: , if n odd. , if n even.
Proof
The graph of is of the form shown in Fig. 1. This means that there exists a permutation matrix P, so that the product will have a block diagonal form, where the diagonal elements are the same with shift circulant basic matrix Cn and where
The shift circulant matrix A6,2
Let the shift circulant matrixWe denote , where , with . We have sets α and products . So, there exist products of the form , with . From these products, +66 are different from zero and are distributed as follows:
- •
There are +6 products, , of the form
- •
There are +36 products, , of the forms
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