The articles [11], [14], [1], [4], [2], [15], [6], [10], [9], [3], [8], [7], [13], [12], and [5] provide the terminology and notation for this paper. The following two propositions are true: (1) 1 = 1CF . (2) 0CF = 0. Let A be a matrix over C. The functor ACF yields a matrix over CF and is defined by: (Def. 1) ACF = A. Let A be a matrix over CF. The functor AC yielding a matrix over C is define...

An n × n complex sign pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is f(λ). If S is a spectrally arbitrary complex sign pattern matrix, and no proper subpattern of S is spectrally arbitrary, then S is a minimal sp...

The eigenvalue densities of complex central Wishart matrices are investigated with the objective of studying an open problem in channel capacity. These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. The connection between the complex Wishart matrix theory and information theory is given. This facil...

Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2, . . . , 16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12, 14 and 16, w...

where Sn is the symmetric group of permutations of the set {1, . . . , n}. The problem of efficient computation of the permanent has attracted a lot of attention. It is #P -hard already for 0-1 matrices [Va79], but a fully polynomial randomized approximation scheme, based on the Markov Chain Monte Carlo approach, is constructed for all non-negative matrices [J+04]. A deterministic polynomial ti...

In the mathematical modeling of real-life applications, systems of equations with complex coefficients often arise. While many techniques of numerical linear algebra, e.g., Krylovsubspace methods, extend directly to the case of complex-valued matrices, some of the most effective preconditioning techniques and linear solvers are limited to the real-valued case. Here, we consider the extension of...

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to th...

In [F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2 case, Linear Algebra Appl., 38:239–249, 1981], the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig’s formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with ran...

This note describes a set of routines for the eigenvalue decomposition, singular value decomposition, and Takagi factorization of a complex matrix. Unlike many other implementations, the current ones are all based on the Jacobi algorithm, which makes the code very compact but suitable only for small to medium-sized problems. Although distributed as a library, the routines are self-contained and...