On Circulant Matrices Irwin Kra and Santiago R

On Circulant Matrices Irwin Kra and Santiago R. Simanca ome mathematical topics—circulant ma- years, the authors’ interest was rekindled by the trices, in particular—are pure gems that casual discovery by S. Simanca that these matrices cry out to be admired and studied with are connected with algebraic geometry over the different techniques or perspectives in mythical field of one element. mind. Circulant matrices are prevalent in many parts SOur work on this subject was originally moti- of mathematics (see, for example, [8]). We point vated by the apparent need of the first author to the reader to the elegant treatment given in [4, derive a specific result, in the spirit of Proposition §5.2] and to the monograph [1] devoted to the 24, to be applied in his investigation of theta subject. These matrices appear naturally in areas constant identities [9]. Although progress on that of mathematics where the roots of unity play a front eliminated the need for such a theorem, role, and some of the reasons for this will unfurl the search for it continued and was stimulated in our presentation. However ubiquitous they are, by enlightening conversations with Yum-Tong Siu many facts about these matrices can be proven during a visit to Vietnam. Upon the first author’s using only basic linear algebra. This makes the return to the U.S., a visit by Paul Fuhrmann brought area quite accessible to undergraduates looking to his attention a vast literature on the subject, for research problems or mathematics teachers including the monograph [4]. Conversations in searching for topics of unique interest to present the Stony Brook mathematics common room at- to their students. tracted the attention of the second author and We concentrate on the discussion of necessary that of Sorin Popescu and Daryl Geller∗ to the and sufficient conditions for circulant matrices to subject and made it apparent that circulant matri- be nonsingular and on various distinct represen- ces are worth studying in their own right, in part tations they have, goals that allow us to lay out because of the rich literature on the subject con- the rich mathematical structure that surrounds necting it to diverse parts of mathematics. These them. Our treatment, though, is by no means ex- productive interchanges between the participants haustive. We expand on their connection to the resulted in [5], the basis for this article. After that algebraic geometry over a field with one element, version of the paper lay dormant for a number of to normal curves, and to Toeplitz’s operators. Irwin Kra is emeritus professor of mathematics at State The latter material illustrates the strong presence University of New York at Stony Brook. His email address these matrices have in various parts of modern is [email protected]. and classical mathematics. Additional connections Santiago R. Simanca is a visiting senior professor of to other mathematics may be found in [11]. mathematics at the Laboratoire Jean Leray, Nantes, The paper is organized as follows. In the France. His email address is [email protected]. section “Basic Properties” we introduce the ba- ∗Our colleague Daryl died on February 5, 2011. We dedi- sic definitions and present two models of the cate this manuscript to his memory. space of circulant matrices, including that as a finite-dimensional commutative algebra. Their de- DOI: http://dx.doi.org/10.1090/noti804 terminant and eigenvalues, as well as some of 368 Notices of the AMS Volume 59, Number 3 th k 1 their other invariants, are computed in the section operator acting on v; its k row is T − v, k = “Determinants and Eigenvalues”. In the section 1,...,n: “The Space of Circulant Matrices” we discuss fur- v0 v1 vn 2 vn 1 ther the space of such matrices and present their · · · − − vn 1 v0 vn 3 vn 2 third model identifying them with the space of − · · · − − . .. diagonal matrices. In the section “Roots of Poly- V . . = nomials” we discuss their use in the solvability of v2 v3 v0 v1 · · · polynomial equations. All of this material is well v1 v2 vn 1 v0 · · · − known. Not so readily found in the literature is We denote by Circ(n) Mn the set of all n n the remaining material, which is also less elemen- complex circulant matrices.⊂ × tary. In the section “Singular Circulant Matrices” It is obvious that Circ(n) is an n-dimensional we determine necessary and sufficient conditions complex vector space (the matrix V is identified for classes of circulant matrices to be nonsingu- with its first row) under the usual operations of lar. The geometry of the affine variety defined by matrix addition and multiplication of matrices these matrices is discussed in the section “The by scalars; hence our first model for circulant Geometry of Circ(n)”, where we also speculate on matrices is provided by the C-linear isomorphism some fascinating connections. In the section “The (FIRST MODEL) I : Circ(n) Cn, Rational Normal Curves Connection” we establish → a relationship between the determinant of a cir- where I sends a matrix to its first row. Matrices culant matrix and the rational normal curve in can, of course, be multiplied, and one can easily complex projective space and uncover their con- check that the product of two circulant matrices nection to Hankel matrices. Finally, in the section is again circulant and that for this set of matrices, “Other Connections—Toeplitz Operators” we re- multiplication is commutative. However, we will late them to the much-studied Toeplitz operators shortly see much more and conclude that we are and Toeplitz matrices as we outline their use in dealing with a mathematical gem. Before that we an elementary proof of Szegö’s theorem. record some basic facts about complex Euclidean It is a pleasure for the first author to thank Yum- space that we will use. Tong Siu for outlining another elementary proof of The ordered n-tuples of complex numbers can formula (3) and for generating his interest in this be viewed as the elements of the inner product n topic. He also thanks Paul Fuhrmann for bringing space C with its Euclidean (L2-norm) and standard to his attention a number of references on the orthonormal basis subject and for the helpful criticism of an earlier ei (δi,0,...,δi,n 1) , i 0,...,n 1, draft of this manuscript. It is with equal pleasure = − = − where δi,j is the Kronecker delta ( 1 for i j and that the second author thanks A. Buium for many = = conversations about the subject of the field with 0 for i ≠ j). We will denote this basis by e and one element and for the long list of related topics remind the reader that in the usual representation n 1 that he brought to his attention. v (v0, v1,...,vn 1) i −0 viei , the vi ’s are the = − = = components of v with respect to the basis e. Basic Properties To explore another basis, we fix once and for all a choice of a primitive nth root of unity We fix hereafter a positive integer n 2. Our ≥ 2πı main actors are the n-dimensional complex vector ǫ e n , space Cn and the ring of n n complex matrices = × define for l 0, 1,...,n 1, Mn. We will be studying the multiplication Mv of = − n matrices M Mn by vectors v C . In this regard, 1 l 2l (n 1)l n ∈ ∈ xl (1, ǫ , ǫ ,...,ǫ − ) C , we view v as a column vector. However, at times, = √n ∈ it is useful mathematically and more convenient and introduce a special case of the Vandermonde typographically to consider matrix v (v0, v1,...,vn 1) Cn 1 1 1 1 = − ∈ · · · n 2 n 1 1 ǫ ǫ − ǫ − n · · · as a row vector. We define a shift operator T : C 1 . Cn → E . .. by = √n n 2 (n 2)2 (n 2)(n 1) 1 ǫ − ǫ − ǫ − − T (v0,v1,...,vn 1) (vn 1,v0,...,vn 2). n 1 · · · (n 1)(n 2) (n 1)2 − = − − 1 ǫ − ǫ − − ǫ − · · · We start with the basic and key definition. It is well known and established by a calculation Definition 1. The circulant matrix V circ v as- that n = { } sociated to the vector v C is the n n matrix n j i ∈ × (1) det E n− 2 (ǫ ǫ ) ≠ 0; whose rows are given by iterations of the shift = 0 i<j n 1 − ≤ ≤ − March 2012 Notices of the AMS 369 hence E is nonsingular. In fact, E is a most Remark 5. The algebra C[X]/(Xn 1) can be iden- 1 t − remarkable matrix: It is unitary, E− E , and it tified with the space Pn 1 of complex polynomials − t 1 = is symmetric, E E, and hence E− E; and its of degree (n 1) with appropriate definition of = = ≤ − columns and rows are the vectors xl . multiplication of its elements. Under this identifi- We view E as a self-map of Cn{ and} conclude cation, for V Circ(n), ∈ that Eei E(ei ) xi . Since E is nonsingular, we J(V) PV (X). = = = see that the xl are another orthonormal basis n { } for C , to be denoted by x. The C-linear self-map Determinants and Eigenvalues n of C defined by the matrix E depends, of course, The Basic Theorem on the bases for the domain and target; to show Many results about circulants follow from direct this dependence, the map should be denoted as calculations; in particular, the next theorem. Ee,e. Observe that as linear maps, Ee,e Ie,x, where = n I is the n n identity matrix.

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