On the matrix equation Am = λJ

In this paper, we will study the h-circulants which satisfy the matrix equation A^m = λJ of n × n matrices. We will determine all the (0, 1) h-circulant solutions for 0 < h < n so that h^m ≡ 0 (mod n).     

Permanental Polytopes of Doubly Stochastic Matrices

A subpolytope Γ of the polytope Ω_n of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω₂ is contained in any permanental polytope of Ω₃ with positive dimension. There is no 3-dimensional permanental polytope of Ω₃, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω₃ (a square of side $\text{1}\text{3}$). Permanental polytopes of dimension $\text{(n}{}^2\text{?3n+4)}\text{2}$ are exhibited for each n?4.     

Szeg? type polynomials and para-orthogonal polynomials

Szeg? type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n?0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szeg? type polynomials are also given. Examples are provided to justify the results.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

On generalized circulants over a Boolean algebra

For any given positive integer n, we give a necessary and sufficient condition for an n × nr-circulant (generalized circulant) over the Boolean algebra B = {0, 1} to be idempotent, and we present an algorith to obtain all n × n indempotent r-circulants over B for r = 0, 1,…,n ? 1.     

Quadratically normal matrices of type 1 and unitary similarities between them

Verification of the unitary similarity between matrices having quadratic minimal polynomials is known to be much cheaper than the use of the general Specht-Pearcy criterion. Such an economy is possible due to the following fact: n × n matrices A and B with quadratic minimal polynomials are unitarily similar if and only if A and B have the same eigenvalues and the same singular values. It is shown that this fact is also valid for a subclass of matrices with cubic minimal polynomials, namely, quadratically normal matrices of type 1.     

Polynomials with respect to a general basis. I. Theory

As n × n Hessenberg matrix A is defined whose characteristic polynomial is relative to an arbitrary basis. This generalizes the companion, colleague, and comrade matrices when the bases are, respectively, power, Chebyshev, and orthogonal, so the term “confederate” matrix is suggested. Some properties of A are derived, including an algorithm for computing powers of A. A scheme is given for inverting the transformation matrix between the arbitrary and power bases. A Vandermonde-type matrix associated with A and a block confederate matrix are defined.     

Multiple roots of diagonal multiples of a square matrix

We find conditions on an n-square matrix A, over a field F of characteristic ≠2, that are equivalent to the following property: for any n-diagonal D over F, the matrix DA has a multiple eigenvalue (or multiple permanental root). Further results of a combinatorial flavour are given in the same direction. We also prove a new criterion for the irreducibility of square matrices.     

Bezoutians and projections

With each polynomial p of degree n whose roots lie inside the unit disc we may associate the n-dimensional space of all solutions of the recurrence relation whose coefficients are those of p (considered as a subspace of 1²). The main result consists in establishing a close relation between the Bezoutian of two such polynomials (of the same degree) and the projection operator onto one of the corresponding spaces along the complement of the other. The note forms a loose continuation of the author's investigations of the infinite companion matrix—the generating function of the infinite companion matrix of a polynomial p appears thus as a particular case; the corresponding Bezoutian is that of the pair p and zⁿ.     

On the number of P-vertices of some graphs

We provide positive answers to some open questions presented recently by Kim and Shader on a continuity-like property of the P-vertices of nonsingular matrices whose graph is a path. A criterion for matrices associated with more general trees to have at most n − 1 P-vertices is established. The cases of the cycles and stars are also analyzed. Several algorithms for generating matrices with a given number of P-vertices are proposed.     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n×n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over Fq of given degrees and for the number of n×n Hankel matrices over Fq of a given rank.     

Congenial matrices

For a given polynomial in the usual power form, its associated companion matrix can be applied to investigate qualitative properties, such as the location of the roots of the polynomial relative to regions of the complex plane, or to determine the greatest common divisor of a set of polynomials. If the polynomial is in “generalized” form, i.e. expressed relative to an orthogonal basis, then an analogue to the companion matrix has been termed the comrade form. This followed a special case when the basis is Chebyshev, for which the term colleague matrix had been introduced. When a yet more general basis is used, the corresponding matrix has been named confederate. These constitute the class of congenial matrices, which allow polynomials to be studied relative to an appropriate basis. Block-partitioned forms relate to polynomial matrices.     

Permanental polynomials of graphs

The first section surveys recent results on the permanental polynomial of a square matrix A, i.e., per(xI – A). The second section concerns the permanental polynomial of the adjacency matrix of a graph. The final section is an introduction to the permanental polynomial of the Laplacian matrix of a graph. An appendix lists some of these latter polynomials.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

Lyapunov,Bézout,and Hankel

Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov-type equations of the form X?SXC=>W and X?CXS=W. We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.     

Permanental pairs of doubly stochastic matrices. II

The n ×n doubly stochastic matrices A, B form a permanental pair if the permanent of every convex linear combination λA+(1?λ)B(0?λ?1) is independent of λ A, B are called mates. In this article we show that the direct sum of any number, k, of matrices J_i (of varying individual dimension) cannot have a mate. Here J_i is the n_i×n_i matrix with every entry equal to $\text{1}\text{n}{}_{\mathrm{i}}$;∑n_i=n.     

Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi β-ensembles of N×N random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N→∞. In the bulk of the spectrum of each β-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_{1}F_{1}$ , whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre β-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when β is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson’s lemma and the steepest descent method for integrals of Selberg type.     

Clusters of point matrices

Along with classical orthogonal polynomials, we consider orthogonal polynomials of degree n ? 1 at n points. These arise naturally from interpolation polynomials. The name “point matrices” is justified by the fact that we deal, not with a class of similar or congruent matrices that play a key role in a linear space and are related to its bases, but with matrices with a fixed set of nodes (or points) x ₁, …, x _n . A certain matrix cluster corresponds to each set of nodes. It is stated that there exists a simple connection between eigenproblems of a Hankel matrix H and a symmetric Jacobi matrix T.     

On convertible nonnegative matrices * †

An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H°A) where H ° Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of

nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.