Roots and canonical forms for circulant matrices

Abstract

Introduction. A square matrix is called circulanti1) if each row after the first is obtained from its predecessor by a cyclic shift. Circulant matrices arise in the study of periodic or multiply symmetric dynamical systems. In particular they have application in the theory of crystal structure [1]. The history of circulant matrices is a long one. In this paper a (block-diagonal) canonical form for circulant matrices is derived. The matrix which transforms a circulant matrix to canonical form is given explicitly. Thus the characteristic roots and vectors of the original circulant can be found by solving matrices of lower order. If the cyclic shift defining the circulant is a shift by one column(2) to the right, the circulant is called simple. Many of the theorems demonstrated here are well known for simple circulants. The theory has been extended to general circulant and composite circulant matrices by B. Friedman [3]. The present proofs are different from his; some of the results obtained go beyond his work. 2. Notations. Definition 2.1. A g-circulant matrix is an nxn square matrix of complex numbers, in which each row iexcept the first) is obtained from the preceding row by shifting the elements cyclically g columns to the right. This connection between the elements afJ-of the ¿th row and the elements of the preceding row is repeated in the formula (2.1) atJ = ai_liJ_f, where indices are reduced to their least positive remainders modulo n. If equation (2.1) holds for all values of i greater than 1, it will hold automatically for i = 1. It is possible to generalize the methods and results of this paper by allowing the elements au of the circulant matrix to be square matrices themselves, all of fixed dimension. This extension is outlined in §6 below. Let A he an arbitrary matrix. If there is a nonzero vector x and a scalar X such that the relation Presented to the Society April 23, 1960 under the title Circulant and composite circulant matrices; received by the editors November 13, 1961. (!) Rutherford [5] uses the term continuant for circulant. (2) See the example of a 5-circulant on p. 31.

Characterization and properties of (Pσ, Q) symmetric and co-symmetric matrices

“…We partition P − and Q − as In [18] we de ned A ∈ C m×n to be (R, Sσ) commutative if RA = ASσ. Much research has been devoted to matrices with this property for speci c choices of R, S, and σ (e.g., [1]- [17], [19]- [21]), the common theme (usually not stated explicitly) being that if RA = ASσ, then P − AQ has a convenient block structure. In all cases that we know of, {λ , λ , .…”

On Moore bipartite digraphs

“…Thus, it is known that the bound Mðd 1 ; d 2 ; kÞ is only attained when 2 k 4, if d 1 d 2 > 1. The non-existence of a Moore bipartite digraph G ¼ ðV 1 [ V 2 ; EÞ with diameter k > 4, out-degrees ðd 1 ; d 2 Þ, d 1 d 2 > 1, and order N ¼ n 1 þ n 2 ,…”

“…where the polynomial arithmetic is modulo x n À 1; see [1]. Moreover, if A is non-singular and has order n, then g has an inverse in…”

“…; d 1 À 1g and A 1 is circulant. Nevertheless, in the case of the digraph 1 , where the greatest integer of N þ ð0Þ is 1 À 1þ 1 2 ð 1 À 1Þ < 1 2 1 ðn À 1Þ=2 (since 2 > 1). Hence, there does not exist an isomorphism between both digraphs that identify the vertices v ¼ 0 of the partite set V 1 of each digraph, which implies that they are not isomorphic, if…”

Zolotarev’s theorem on the Legendre symbol

“…I thank Professor D. H. Lehmer for asking whether the methods developed in [1] could be used to prove that det A(a) = (a/p).…”