In this paper we investigate the question of convergence for degenerate descent procedures. The results focus attention on the role of a property of symmetric matrices analogous to, but different from, the usual condition number used in numerical linear algebra. The main result is a bound which establishes a specific rate of exponential decay for time varying linear systems with singular A matr...

A new algebraic framework is introduced based on stochastic matrices. Together with several new operators, the set of stochastic matrices is shown to constitute a vector space, an inner-product space, and an associative algebra. The new zero vector is the uniform probability distribution and linear addition is akin to statistical independence. This new stochastic algebra allows Markov chains an...

A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro’s “Pfister Factor Conjecture” is given for tensor products of at most five quaternion algebras. Throughout this paper, the characteristic of the base...

The question which multivariate GARCH models in the vec form are representable in the BEKK form is addressed. Using results from linear algebra, it is established that all vec models not representable in the simplest BEKK form contain matrices as parameters which map the vectorised positive semi-definite matrices into a strict subset of themselves. Moreover, a general result from linear algebra...

It is shown that every  almost linear bijection $h : Arightarrow B$ of a unital $C^*$-algebra $A$ onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$-algebra $A$ of real rank zero onto a unital$C^*$-algebra...

In this paper, we study the stochastic root matrices of stochastic matrices. All stochastic roots of 2¥2 stochastic matrices are found explicitly. A method based on characteristic polynomial of matrix is developed to find all real root matrices that are functions of the original 3¥3 matrix, including all possible (function) stochastic root matrices. In addition, we comment on some numerical met...

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifold...

A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined “quadquaternion” signal and noise...

In the age of statistical and scientific databases, there is an emerging trend of integrating analytical algorithms into database systems. Many of these algorithms are based on linear algebra with large, sparse matrices. However, linear algebra expressions often contain multiplications of more then two matrices. The execution of sparse matrix chains is nontrivial, since the runtime depends on t...

Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well known theory of diagonalization for symmetric matrices makes the topic immediately accessible to...