A complete unitary similarity invariant for unicellular matrices

We present necessary and sufficient conditions for an n×n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n×n complex matrix A.     

On the completion of a partial integral matrix to a unimodular matrix

We prove that if a partial integral matrix has a free diagonal then this matrix can be completed to a unimodular matrix. Such a condition is necessary in a general sense. Consequently if an n × n (n ? 2) partial integral matrix has 2n − 3 prescribed entries and any n entries of these do not constitute a row or a column then it can be completed to a unimodular matrix. This improves a recent result of Zhan.     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

Possible spectra of totally positive matrices

For any given set S of n distinct positive numbers, we construct a symmetric n-by-n (strictly) totally positive matrix whose spectrum is S. Thus, in order to be the spectrum of an n-by-n totally positive matrix, it is necessary and sufficient that n numbers be positive and distinct.     

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.     

Products of irreducible matrices

In this paper we characterize the subsemigroup of B_n (B_n is the multiplicative semigroup of n × n Boolean matrices) generated by all the irreducible matrices, and hence give a necessary and sufficient condition for a Boolean matrix A to be a product of irreducible Boolean matrices. We also give a necessary and sufficient condition for an n × n nonnegative matrix to be a product of nonnegative irreducible matrices.     

Wishartness and independence of matrix quadratic forms in a normal random matrix

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y. The general covariance Σ_Y of Y means that the collection of all np elements in Y has an arbitrary np×np covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y^′W_iY's with the symmetric W_i's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y^′W_iY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.     

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.     

Matrices permutation equivalent to primitive matrices

We give a necessary and sufficient condition for an n×n (0,1) matrix (or more generally, an n×n nonnegative matrix) to be permutation equivalent to a primitive matrix. More precisely, except for two simple permutation equivalent classes of n×n (0,1) matrices, each n×n (0,1) matrix having no zero row or zero column is permutation equivalent to some primitive matrix. As an application, we use this result to characterize the subsemigroup of B_n (B_n is the multiplicative semigroup of n×n Boolean matrices) generated by all the primitive matrices and permutation matrices. We also consider a more general problem and give a necessary and sufficient condition for an n×n nonnegative matrix to be permutation equivalent to an irreducible matrix with given imprimitive index.     

Positive definiteness of Hermitian interval matrices

We present a new necessary and sufficient criterion to check the positive definiteness of Hermitian interval matrices. It is shown that an n×n Hermitian interval matrix is positive definite if and only if its 4^n-1(n-1)! specially chosen Hermitian vertex matrices are positive definite.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Irreducible Restrictions of Representations of the Symmetric Groups

A proof is given of a recent conjecture of Jantzen and Seitzgiving a necessary and sufficient condition for a representationof the symmetric group on n objects (over an algebraically closedfield of prime characteristic p < n) to remain irreducibleupon restriction to the symmetric group on n–1 objects.     

Counting extreme <Emphasis Type="Italic">U</Emphasis><Subscript>1</Subscript> matrices and characterizing quadratic doubly stochastic operators

The U ₁ matrix and extreme U ₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U ₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U ₁ matrix and investigated the structure of extreme U ₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U ₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.     

The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix

Let W_n be n×n Hermitian whose entries on and above the diagonal are independent complex random variables satisfying the Lindeberg type condition. Let T_n be n×n nonnegative definitive and be independent of W_n. Assume that almost surely, as n→∞, the empirical distribution of the eigenvalues of T_n converges weakly to a non-random probability distribution.Let . Then with the aid of the Stieltjes transforms, we show that almost surely, as n→∞, the empirical distribution of the eigenvalues of A_n also converges weakly to a non-random probability distribution, a system of two equations determining the Stieltjes transform of the limiting distribution. Important analytic properties of this limiting spectral distribution are then derived by means of those equations. It is shown that the limiting spectral distribution is continuously differentiable everywhere on the real line except only at the origin and that a necessary and sufficient condition is available for determining its support. At the end, the density function of the limiting spectral distribution is calculated for two important cases of T_n, when T_n is a sample covariance matrix and when T_n is the inverse of a sample covariance matrix.     

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.     

Eigenvalue location using certain matrix functions and geometric curves

The main inertia theorem gives necessary and sufficient conditions that an n×n complex matrix A have no eigenvalues on the imaginary axis of the complex plane. In this paper corresponding necessary and sufficient conditions are given that A have no eigenvalues on any arbitrary circle, line, and certain other curves in the complex plane. Generalizations of the second part of the main inertia theorem give inclusion regions for the eigenvalues of A.     

A necessary and sufficient condition for the nirenberg problem

We seek metrics conformal to the standard ones on Sⁿ having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S², for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.     

The group marriage problem

Let G be a permutation group acting on [n]={1,…,n} and V={Vi:i=1,…,n} be a system of n subsets of [n]. When is there an element g∈G so that g(i)∈Vi for each i∈[n]? If such a g exists, we say that G has a G-marriage subject to V. An obvious necessary condition is the orbit condition: for any nonempty subset Y of [n], there is an element g∈G such that the image of Y under g is contained in _?y∈YVy. Keevash observed that the orbit condition is sufficient when G is the symmetric group Sn; this is in fact equivalent to the celebrated Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and only if G is a direct product of symmetric groups. We extend the notion of orbit condition to that of k-orbit condition and prove that if G is the cyclic group Cn where n?4 or G acts 2-transitively on [n], then G satisfies the (n−1)-orbit condition subject to V if and only if G has a G-marriage subject to V.