Invariant functions of the elements of the powers of a square matrix

A square matrix A is raised to any real power n, negative or fractional values being permitted when Aⁿ can be defined; C is a matrix that commute with A. Linear identities existing between the elements of Aⁿ or C are investigated, in such a way that the number of elements in each identity is minimized in general. Both this number and the method of investigation depend on the Jordan canonical form of A, but if A has a special property, some of these identities and their method of derivation are independent of the Jordan canonical form.     

Essentially doubly stochastic matrices II. Canonical forms for row equivalence,equivalence, and a special case of congruence

In this paper we obtain canonical forms for row equivalence, equivalence, and a special case of congruence in the algebra of essentially doubly stochastic (e.d.s.) matrices of order n over a field F, char(F) [nmid] n. These forms are analogues of familiar forms in ordinary matrix algebra. The canonical form for equivalence is used in showing, in a subsequent paper, that every e.d.s. matrix of rank r and order n over F, char(F) = 0 or char(F) > n, is a product of elementary e.d.s. matrices, n – r of which are singular.     

Computing the determinantal representations of hyperbolic forms

The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0, 1, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.     

A matrix pencil approach computing the elementary divisors of a matrix: Numerical aspects and applications

In the present paper is presented a new matrix pencil-based numerical approach achieving the computation of the elementary divisors of a given matrixA ∈ C ^{n × n}. This computation is attained without performing similarity transformations and the whole procedure is based on the construction of the Piecewise Arithmetic Progression Sequence (PAPS) of the associated pencil λI _n - A of matrix A, for all the appropriate values of λ belonging to the set of eigenvalues of A. This technique produces a stable and accurate numerical algorithm working satisfactorily for matrices with a well defined eigenstructure. The whole technique can be applied for the computation of the first, second and Jordan canonical form of a given matrixA ∈ C ^{n × n}. The results are accurate for matrices possessing a well defined canonical form. In case of defective matrices, indications of the most appropriately computed canonical form are given.     

On the accuracy of the Gerschgorin circle theorem for bounding the spread of a real symmetric matrix

The spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues. The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread. For nonsymmetric matrices the Gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable. This is not true for real symmetric matrices. It is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by $\text{p+1}$, where p is the maximum number of off diagonal nonzeros in any row of the matrix. For full matrices this is just $\text{n}$. This bound is not quite sharp for n greater than 2, but examples with ratios of $\text{n?1}$ for all n are given. For banded matrices with m nonzero bands the maximum ratio is bounded by $\text{m}$ independent of the size of n. This bound is sharp provided only that n is at least 2m. For sparse matrices, p may be quite small and the Gerschgorin bound may be surprisingly accurate.     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

On the similarity of matrices of even order

We study invariants of simultaneous similarity of a pair of matrices of even order 2k over the field of complex numbers for the case in which all elementary divisors of the corresponding characteristic polynomial matrix are identical and their number is k.     

Transposing a matrix by similarity operations

A matrix T is said to co-transpose a square matrix A if T^?1AT=A′ and T^?1A′T=A. For every n?3 there exists a real n×n matrix which cannot be co-transposed by any matrix. However, it is shown that the following classes of real matrices can be co-transposed by a symmetric matrix of order two: 2×2 matrices, normal matrices, and matrices whose square is symmetric.     

Systems of second-order linear ODE’s with constant coefficients and their symmetries

Starting from the study of the symmetries of systems of 4 second-order linear ODEs with constant real coefficients, we determine the dimension and generators of the symmetry algebra for systems of n equations described by a diagonal Jordan canonical form. We further prove that some dimensions between the lower and upper bounds cannot be attained in the diagonal case, and classify the Levi factors of the symmetry algebras.     

Involutory matrices over finite commutative rings

An n × n matrix A is called involutory iff A²=I_n, where I_n is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set $\text{C}$ with respect to similarity for the n × n involutory matrices over R—i.e., a set $\text{C}$ of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in $\text{C}$. Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set $\text{C}$ for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A₁,…,A_v of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each A_i). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2^m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

Canonical forms of some special matrices useful in statistics

In experimental situations wheren two or three level factors are involved andn observations are taken, then theD-optimal first order saturated design is ann ×n matrix with elements ±1 or 0, ±1 with the maximum determinant. Canonical forms are useful for the specification of the non-isomorphicD-optimal designs. In this paper, we study canonical forms such as the Smith normal form, the first, second and the Jordan canonical form ofD-optimal designs. Numerical algorithms for the computation of these forms are described and some numerical examples are also given.     

Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution

It is well known that if P is a nonnegative matrix, then its spectral radius is an eigenvalue of P (Perron-Frobenius theorem). In this paper it is shown that if P is an n × n nonnegative matrix and it commutes with a nonnegative symmetric involution when n=4m+3, then (1) P has at least two real eigenvalues if n=4m or 4m + 2, (2) P has at least one real eigenvalue if n=4m+1, and (3) P has at least three real eigenvalues if n=4m+3, where m is a nonnegative integer and n ? 1. Examples are given to show that these results are the best possible, and nonnegative symmetric involutions are classified.     

Tight bounds on the spectral radius of asymmetric nonnegative matrices

For an arbitrary asymmetric nonnegative n × n matrix A we identify a pair of symmetric matrices whose largest eigenvalues bound the spectral radius of A. Furthermore, we show that these bounding matrices are best possible by characterizing matrices A which attain equality with either the upper or the lower bounding matrix. The lower bound may be extended to some matrices with negative entries provided they have no negative cycles.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

Solutions to an inverse monic quadratic eigenvalue problem

Given n+1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n×n so that the quadratic pencil Q(λ)=λ²I_n+λC+K has the given n+1 pairs as eigenpairs. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained. Numerical algorithms for solving the problem are developed. Numerical examples illustrating these solutions are presented.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

On fractional Hadamard powers of positive definite matrices

Let A = (a_ij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that A^α ≡ (a_ij^α) is positive definite for all real α ? n ? 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix.