The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n n random matrices within spectral intervals of the order O(n ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermi...

If k = 1, then X is called the group inverse of A, and is denoted by X = Ag. The Drazin inverse is very useful in various applications (see, e.g. [1]–[4]; applications in singular differential and difference equations, Markov chains and iterative methods). In 1980, Cline and Greville [5] extended the Drazin inverse of square matrix to rectangular matrix, which can be generalized to the quaterni...

If (without loss of generality) n = p, where p is prime, divide and conquer Fourier transforms using O(nlogn) operations reduce multiplying, or inverting nonsingular, complex n×n matrices to abelian group algebra convolutions. If M is a complex 2×2 matrix, constructing a unitary matrix T and an upper triangular matrix TMT reduces to n(n−1)/2 such constructions in which a 2×2 matrix μ is transfo...

Based on crossed-dipole antenna arrays, quaternion-valued data models have been developed for both direction of arrival estimation and beamforming in the past. However, for almost all the models, and especially for adaptive beamforming, the desired signal is still complex-valued as in the quaternion-valued Capon beamformer. Since the complex-valued desired signal only has two components, while ...

A common measure of conformational similarity in structural bioinformatics is the minimum root mean square deviation (RMSD) between the coordinates of two macromolecules. In many applications, the rotations relating the structures are not needed. Several common algorithms for calculating RMSDs require the computationally costly procedures of determining either the eigen decomposition or matrix ...

For spacecraft attitude representation, the quaternion is preferred over other representations due to its bilinear nature of the kinematics and the singularity-free property. However, when applied in the attitude estimation filters, its unity norm constraint can be easily destroyed by the quaternions averaging operation. Although the brute-force quaternion normalization can be implemented, this...

As is well-known, the real quaternion division algebra H is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra O can not be algebraically isomorphic to any matrix algebras over the real number field R, because O is a non-associative algebra over R. However since O is an extension of H by the Cayley-Dickson process and is also finite-dimensional, som...

This paper provides a general construction technique for rectangular matrices whose elements are quaternion variables and whose column vectors are formally orthogonal. These matrices are named quaternion orthogonal designs, in parallel with the well-known constructs of real and complex orthogonal designs. The proposed construction technique provides the first infinite family of this type of qua...

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...

In this paper, we address the problem of restoring orthogonality a numerically noisy 4D rotation matrix by finding its nearest (in Frobenius norm) correct matrix. This can be straightforwardly solved using Singular Value Decomposition (SVD). Nevertheless, to avoid numerical methods, present two new closed-form methods. One relies on direct minimization mentioned norm, and other passage double q...