Stochastic aspects of the unitary dual group

In this note, we study the asymptotic properties of the ?-distribution of traces of some matrices, with respect to the free Haar trace on the unitary dual group. The considered matrices are powers of the unitary matrix generating the Brown algebra. We proceed in two steps, first computing the free cumulants of any R-cyclic family, then characterizing the asymptotic ?-distributions of the traces of powers of the generating matrix, thanks to these free cumulants. In particular, we obtain that these traces are asymptotic ?-free circular variables.     

A method for determining the partial indices of symmetric matrix functions

We propose a method for determining the partial indices of matrix functions with some symmetries. It rests on the canonical factorization criteria of the author’s previous articles. We show that the method is efficient for the symmetric classes of matrix functions: unitary, hermitian, orthogonal, circular, symmetric, and others. We apply one of our results on the partial indices of Hermitian matrix functions and find effective well-posedness conditions for a generalized scalar Riemann problem (the Markushevich problem).     

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew‐Hermitian logarithm of a unitary matrix and also skew‐Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near ? 1, lead to very non‐Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew‐Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J‐skew‐symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Conjugate-normal matrices with conjugate-normal submatrices

In studying the reduction of a complex n × n matrix A to its Hessenberg form by the Arnoldi algorithm, T. Huckle discovered that an irreducible Hessenberg normal matrix with a normal leading principal m × m submatrix, where 1 < m < n, actually is tridiagonal. We prove a similar assertion for the conjugate-normal matrices, which play the same role in the theory of unitary congruences as the conventional normal matrices in the theory of unitary similarities. This fact is stated as a purely matrix-theoretic theorem, without any reference to Arnoldi-like algorithms. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 21–25.     

On unitary transposable matrices of order three

Mathematical Notes - A matrix A ∈ M n (?) is said to be unitarily transposable if A T = Q*AQ for a certain unitary matrix Q. Every 2 × 2 matrix is unitarily transposable; however,...     

On unitary interpolants and Fredholm infinite block Toeplitz matrices

Conditions for the existence and uniqueness of unitary n × n matrix valued functions f on the unit circle with prescribed Fourier coefficients f_j for j 0 are given (in terms of infinite block Hankel matrices based on the prescribed coefficients f₀,f₁, ) for a natural class of functions. A unitary function belongs to this class if and only if it admits a generalized factorization (in a sense which will be made precise in the paper) or equivalently if and only if any one (and hence both) of the two Toeplitz operators defined by the function are Fredholm. In particular this class includes all continuous unitary n × n matrix valued functions. It is shown that the nonnegative factorization indices of every such unitary f are uniquely determined by f₀,f₁, and formulas for them are given.     

Numerical semigroups generated by primes

Semigroup Forum - Let $$p_1=2, p_2=3, p_3=5, \ldots$$ be the consecutive prime numbers, $$S_n$$ the numerical semigroup generated by the primes not less than $$p_n$$ and $$u_n$$ the largest...     

关于k-广义酉矩阵

本文研究了k-广义酉矩阵的性质及其与酉矩阵、辛矩阵、Householder矩阵之间的联系,取得了许多新的结果,推广了酉矩阵及Householder矩阵的相应结果,特别将正交矩阵的广义Cayley分解推广到了广义酉矩阵上;并将各类酉矩阵及辛矩阵统一了起来.     

广义酉矩阵与广义Hermite矩阵

给出了广义酉矩阵与广义(斜)Hermite矩阵的概念，研究了它们的性质及其与酉阵、共轭辛阵、Hermite阵、Hamilton及广义逆矩阵之间的联系；取得了许多新的结果；推广了酉矩阵、Hermite阵与斜Hermite阵间的相应结果，特别将正交阵的广义Cayley分解推广到了广义酉矩阵上；将各类酉矩阵、Hermite矩阵及广义逆矩阵统一了起来．     

Partial Indices of Unitary and Hermitian Matrix Functions

We show that all partial indices of an even unitary matrix function of size n > 1 are equal to zero. We find new effective sufficient conditions for the existence of canonical factorization of a Hermitian matrix function of size 2 (when all its partial indices are equal to zero).     

共轭正规矩阵的酉相合标准形

本文借助于对称矩阵和反对称矩阵的酉相合标准形讨论了共轭正规矩阵在酉相合下的标准形。     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field Fq2, which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for an m×n matrix A over Fq2 having an M-P inverse are obtained, which make clear the set of m×n matrices over Fq2 having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

An efficient Total Least Squares algorithm based on a rank-revealing two-sided orthogonal decomposition

Solving Total Least Squares (TLS) problemsAXB requires the computation of the noise subspace of the data matrix [A;B]. The widely used tool for doing this is the Singular Value Decomposition (SVD). However, the SVD has the drawback that it is computationally expensive. Therefore, we consider here a different so-called rank-revealing two-sided orthogonal decomposition which decomposes the matrix into a product of a unitary matrix, a triangular matrix and another unitary matrix in such a way that the effective rank of the matrix is obvious and at the same time the noise subspace is exhibited explicity. We show how this decompsition leads to an efficient and reliable TLS algorithm that can be parallelized in an efficient way.     

酉空间中一类子空间的排列问题与具有纠错能力的d~z-析取矩阵

Pooling设计在实践中有着广泛的应用,它的数学模型是d~z-析取矩阵.本文利用酉空间的一类子空间构做了一类新的d~z-析取矩阵.为了讨论此设计的纠错能力,重点研究了酉空间中的一类子空间的排列问题,即对于酉空间F_q~2~((n))上的一个给定的(m,s)型子空间C和一个整数d,找到C的d个(m-1,s-1)型子空间H_1,H_2,…,H_d,使得包含在H_1∪H_2∪…∪H_d中的(r,s-4)型子空间的个数最多,并确定这个数的上界.然后应用此结果,给出了d~z-析取矩阵中反映纠错能力的z值的紧界.     

When is the numerical range of a nilpotent matrix circular?

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces tr M^∗M² and tr M^∗M³ are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5×5 nilpotent matrix has circular numerical range are thoroughly discussed.     

Weighted infinitesimal unitary bialgebras,pre-Lie,matrix algebras,and polynomial algebras

Motivated by comatrix coalgebras, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via the construction of a suitable coproduct. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on matrix algebras. By the close relationship between pre-Lie algebras and infinitesimal unitary bialgebras, we erect a pre-Lie algebra and a new Lie algebra on matrix algebras. Finally, a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras is also given.     

A generalized unitary Hessenberg matrix

This paper describes the use of a generalized isometric Arnoldi algorithm to reduce a unitary matrix, via unitary similarity, to a product of elementary reflectors and permutations. The computation is analogous to the reduction of a unitary matrix to a unitary Hessenberg matrix using the isometric Arnoldi algorithm. In the case in which A is a shift matrix, the reduction provides a novel recurrence for the factor R in the QR factorization of a Toeplitz-like matrix.     

GCD和GCUD矩阵行列式的一个不等式（英）

Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).     

关于次酉矩阵与次镜象矩阵

提出了共轭次转置矩阵、次酉矩阵与次镜象矩阵的概念，对它们的基本性质及其与（反）次Kermite阵的关系进行了深入的研究，获得了一些新的结果，将正交阵的广义Gayley分解推广到了次酉阵上。