Some quaternion matrix equations involving Φ-Hermicity

Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.

The general $\phi$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is the real quaternion algebra. Let $A_{\phi}$ be the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{T}$, where $A\in\mathbb{H}^{m\times n}$ and $\phi$ is a nonstandard involution of $\mathbb{H}$. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix $A_{\phi}$ are given. Two systems of mixed pairs of quaternion matrix Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}$ and $A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2}$ are considered, where $Z$ is $\phi$-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution $(X,Y,Z)$ to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.

An induced real quaternion spherical ensemble of random matrices

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

PROBABILITY DENSITIES AND DISTRIBUTIONS FOR SPIKED AND GENERAL VARIANCE WISHART β-ENSEMBLES

A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue b different from unity. As b increases through b = 2, a gap forms from the largest eigenvalue to the rest of the spectrum, and with b - 2 of order N-1/3 the scaled largest eigenvalues form a well-defined parameter dependent state. Recent works by Bloemendal and Virág [Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825], and Mo [Rank I real Wishart spiked model, Comm. Pure Appl. Math.65 (2012) 1528–1638], have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart β-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart β-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices (β = 4) the latter is recognized as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825]. We also use the construction of the spiked Wishart β-ensemble from [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825] to give a simple derivation of the explicit form of the eigenvalue PDF.