1. Matrices, Vectors and their Basic Operations 1.1. Matrices 1.2. Vectors 1.3. Addition and Scalar Multiplication of Matrices 1.4. Multiplication of Matrices 2. Determinants 2.1. Square Matrices 2.2. Determinants 2.3. Cofactors and the Inverse Matrix 3. Systems of Linear Equations 3.1. Linear Equations 3.2. Cramer’s Rule 3.3. Eigenvalues of a Complex Square Matrix 3.4. Jordan Canonical Form 4....

In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structurepreserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable. 2014 Elsevier Inc. All rights reserved.

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

In this paper, we first discuss the singular value decomposition (SVD) of a quaternion matrix and propose an algorithm to calculate the SVD of a quaternion matrix using its equivalent complex matrix. The singular values of a quaternion matrix are still real and positive, but the two unitary matrices are quaternion matrices with quaternion entries. Then, applications for color image processing b...

Let H be the real quaternion algebra and Hm×n denote set of all m×n matrices over H. For A?Hm×n, we by A? n×m matrix obtained applying ? entrywise to transposed AT, where is a non-standard involution A?Hn×n said ?-skew-Hermicity if A=?A?. In this paper, provide some necessary sufficient conditions for existence ?-skew-Hermitian solution system equations with four unknowns AiXi(Ai)?+BiXi+1(Bi)?=...

The subject of this paper is the Brauer group of a nonsingular complex projective variety. More specifically, we study the question of whether a 2-torsion element of the cohomological Brauer group is representable by a quaternion algebra over the generic point. Using intersection theory – on schemes and on algebraic stacks – we are able to describe an obstruction to such a representation, and t...

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is s...

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is s...