A metric variant of Frobenius's theorem and some other remarks on positive matrices

The theory of positive (=nonnegative) finite square matrices continues, three quarters of a century after the pioneering and well-known papers of Perron and Frobenius [4], to present a multitude of different aspects. This is evidenced, for example, by the recent papers [1] and [2], as well as by the vast literature concerned with extensions to operators on infinite dimensional spaces (see [5]). Supposing A to be a positive n × n matrix with spectral radius r(A) = 1, the main purpose of this note is to display the role of λ = 1 as a root of the minimal polynomial of A (or equivalently, of certain norm conditions on A, for the lattice structure of the space M spanned by the unimodular eigenvectors of A as well as for the permutational character of A on M. Proposition 1 can thus be viewed as a variant of Frobenius's theorem on the peripheral spectrum of indecomposable square matrices, and we hope that the proof of Proposition 2 will clarify to what extent indecomposability is responsible for the main results available in that special case. The remaining remarks (Propositions 3 and 4) are concerned with the spectral characterization of permutation matrices and with finite groups of positive matrices. Some of that material is undoubtedly known, but we give simple, transparent proofs.     

Some remarks on permutation designs

A definition of isomorphism of two permutation designs is proposed, which differs from the definition in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392]. The proposed definition has the (generally required) property that the allowed permutations always transform a permutation design into a permutation design. It is shown that the n permutation designs coming from the partitioning of S_n into permutation designs, as constructed in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392] are all isomorphic. Further we find that this modified definition does not increase the number of nonisomorphic (6, 4) permutation designs. The same investigation showed that one of the designs, claimed to be a (6, 4) permutation design in [J. Combinatorial Theory Ser. A21 (1976), 384–392], is actually not a (6, 4) permutation design.     

Some infinite classes of Hadamard matrices

Some results of Williamson [Duke Math. J., 11 1944, Bull. Amer. Math. Soc., 53 1947] and Wallis (J. Combinatorial Theory, 6 1969] are considerably improved to establish that in each case referred to, the same stated condition or conditions, which according to either of the authors give rise to one Hadamard matrix, actually imply the existence of an infinite series of Hadamard matrices. Also proved is the existence of some infinite series of Williamson's matrices, which coupled with the interesting findings of Turyn [J. Combinatorial Theory Ser. A, 16 1974] establish the existence of infinitely many more series of Hadamard matrices than those known so far.     

Embeddings of minimal non-abelian p-groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.     

On polyhedra of Perron-Frobenius eigenvectors

This paper deals with the positive eigenvectors of nonnegative irreducible matrices which are merely characterized by a given upper bound u on their spectral radius and by a given matrix L of lower bounds for their elements. For any such matrix, the normalized positive left [right] eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows [columns] of the matrix $\text{(u}\text{I}\text{?}\text{L}\text{)}{}^{\mathrm{?1}}$. This polyhedron is proven to be also the smallest closed set which is guaranteed to contain the positive left [right] normalized eigenvector; its vertices are therefore the best componentwise bounds one can obtain on the positive eigenvectors of these matrices. A less general result has also been obtained for the symmetrical case, when the matrices are only characterized by a given lower bound l on their spectral radius and by a given matrix U of upper bounds for their elements.     

Identities among certain triangular matrices

The object of this paper is to develop the ideas introduced in the author's paper [1] on matrices which generate families of polynomials and associated infinite series. A family of infinite one-subdiagonal non-commuting matrices Q_m is defined, and a number of identities among its members are given. The matrix Q₁ is applied to solve a problem concerning the derivative of a family of polynomials, and it is shown that the solution is remarkably similar to a conventional solution employing a scalar generating function. Two sets of infinite triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al. [24-26].     

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free.

In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.     

Invariant scale matrix hypothesis tests under elliptical symmetry

The usual assumption in multivariate hypothesis testing is that the sample consists of n independent, identically distributed Gaussian m-vectors. In this paper this assumption is weakened by considering a class of distributions for which the vector observations are not necessarily either Gaussian or independent. This class contains the elliptically symmetric laws with densities of the form f(X(n × m)) = ψ[tr(X ? M)′ (X ? M)Σ^?1]. For testing the equality of k scale matrices and for the sphericity hypothesis it is shown, by using the structure of the underlying distribution rather than any specific form of the density, that the usual invariant normal-theory tests are exactly robust, for both the null and non-null cases, under this wider class.     

Circular representation problem on hypergraphs

A hypergraph J=(X,E) is said to be circular representable, if its vertices can be placed on a circle, in such way that every edge of H induces an interval. This concept is a translation into the vocabulary of hypergraphs of the circular one's property for the (0, 1) matrices [6] studied by Tucker [9, 10]. We give here a characterization of the hypergraphs which are circular representable. We study when the associated representation is unique, and we characterize the possible transformations of a representation into another, a kind of problem which has already been treated from the algorithmic point of view by Booth and Lueker [1] or Duchet [2] in the case of the interval representable hypergraphs.Finally, we establish a connection between circular graphs and circular representable hypergraphs of the type of the Fulkerson-Gross connection between interval graphs and matrices having the consecutive one's property [5], in some special cases.     

Reed-Muller codes and permutation decoding

We show that the first- and second-order Reed-Muller codes, R(1,m) and R(2,m), can be used for permutation decoding by finding, within the translation group, (m−1)- and (m+1)-PD-sets for R(1,m) for m≥5,6, respectively, and (m−3)-PD-sets for R(2,m) for m≥8. We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967-1970].     

Permutation-matrix groups with positive sum

Explicit forms are given for all commutative sets of permutation matrices which sum to a positive matrix, and for nonabelian groups of permutation matrices of order twice an odd prime which sum to the matrix of all ones. A relationship to circulants of level k is indicated.     

The partition problem for finite abelian groups

By the aid of a slight generalization of the Hales-Jewett theorem [Trans. Amer. Math. Soc.106 (1963), 222–229] we investigate the partition problem for finite Abelian groups. In particular the partition problem for the class of finitely generated free modules over Z_q is solved. By the results of Deuber and Rothschild [“Coll. Math. Soc. János Bolyai 18,” 1976] this yields a complete characterization of those finite Abelian groups with respect to which the class FAB of all finite Abelian groups has the partition property. Especially it turns out that FAB has the partition property with respect to the cyclic group Z_m, m > 1.     

On the asymptotic convergence factor of the total step method in interval computation

For a special class of n×n interval matrices A we derive a necessary and sufficient condition for the asymptotic convergence factor α of the total step method x^(m+1)=Ax^(m)+b to be less than the spectral radius ϱ(|A|) of the absolute value |A| of A.     

On the permutation groups of cyclic codes

We classify the permutation groups of cyclic codes over a finite field. As a special case, we find the permutation groups of non-primitive BCH codes of prime length. In addition, the Sylow p-subgroup of the permutation group is given for many cyclic codes of length p ^m . Several examples are given to illustrate the results.     

A note on preconditional diagonally dominant matrices

This note points out that the main results of [Appl. Math. Comput. 114 (2000) 255] is not true. We show that (1) Theorem 2.1 in [Appl. Math. Comput. 114 (2000) 255] is well known, (2) There are no nonsingular matrices satisfying the sufficient conditions for ensuring diagonally dominance given in Theorem 3.1, and (3) Theorem 4.1 for preconditioning p-cyclic matrices is not true. We also prove that p-cyclic matrices can be column diagonally preconditioned, with a special row permutation if required, to be row diagonally dominant under some assumptions.     

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.     

Riesz potentials and integral geometry in the space of rectangular matrices

Riesz potentials on the space of rectangular n×m matrices arise in diverse “higher rank” problems of harmonic analysis, representation theory, and integral geometry. In the rank-one case m=1 they coincide with the classical operators of Marcel Riesz. We develop new tools and obtain a number of new results for Riesz potentials of functions of matrix argument. The main topics are the Fourier transform technique, representation of Riesz potentials by convolutions with a positive measure supported by submanifolds of matrices of rank<m, the behavior on smooth and L^p functions. The results are applied to investigation of Radon transforms on the space of real rectangular matrices.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).     

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

Decomposition into a direct sum of irreducible representations of the representation of the full collineation group of a finite Desarguesian plane, as a group of matrices permuting the flags of the plane and the simple components of the corresponding commutant algebra, have been worked out here for the projective plane PG(2, 2) and the affine plane EG(2, 3). The dimension and the components of the covariance matrix of the observations from a design derived from such a plane, which commutes with such a permutation representation of the full collineation group of the plane, are thus determined. This paper is in the spirit of earlier works by, James (1957), Mann (1960), 6., 7., McLaren (1963), and Sysoev and Shaikan (1976). A. T. James, Ann. Math. Statist.28 (1957), 993–1002, H. B. Mann, Ann. Math. Statist.31 (1960), 1–15, E. J. Hannan, Research Report (Part. (I)), Summer Research Institute, Australian Math. Soc. and Methuen's Monographs on Applied Probability and Statistics, Supplementary Review Series in Applied Probability, Vol. 3, A. D. McLaren, Proc. Cambridge Philos. Soc.59 (1963), 431–450, and L. P. Sysoev and M. E. Shaikin, Avtomat. i Telemekh.5 (1976), 64–73.