II. LINEAR ALGEBRA Linear Algebra is the study of linear transformations. We used the following result on how to diagonalize an Hermitian matrix: II.1. Theorem: Diagonalization of Hermitian Matrix (Notes on Linear Algebra Theorem 6) Let H ∈ Cn×n be a Hermitian matrix, i.e., such that H = H∗. The eigenvalues λ1, . . . , λn of H are real (they are not necessarily distinct); H has n orthonormal ei...

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $$L_K(\mathcal {G})$$ of an ultragraph $$\mathcal {G}$$ over a field K is purely infinite simple that it von Neumann regular. Consequently, obtain every graded either locally matricial algebra, or full matrix ring $$K[x, x^{-1}]$$ , algebra.

We compute the center in the case where q is a root of unity. Main steps are to compute the degree of an associated quasipolynomial algebra and to compute the dimensions of some interesting irreducible modules. 1. Introduction Quantum groups arose from a quantum mechanical problem in statistical mechanics , the quantum Yang-Baxter equation (QYBE). Following Drinfeld 5], a quantum group is the d...

Matrix Algebra and Deep Neural Networks represent foundational classes of computational algorithms across multiple emerging applications like Augmented Reality or Virtual Reality, autonomous navigation (cars, drones, robots), data science, various artificial intelligence-driven solutions. An accelerator-based architecture can provide performance energy efficiency supporting fixed functions thro...

We examine a natural supersymmetric extension of the bosonic covariant 3-algebra model for M-theory proposed in [1]. It possesses manifest SO(1,10) symmetry and is constructed based on the Lorentzian Lie 3-algebra associated with the U(N) Lie algebra. There is no ghost related to the Lorentzian signature in this model. It is invariant under 64 supersymmetry transformations although the supersym...

Numerical polynomial algebra emerges as a growing field of study in recent years with a broad spectrum of applications and many robust algorithms. Among the challenges in solving polynomial algebra problems with floating-point arithmetic, difficulties frequently arise in regularizing ill-posedness and handling large matrices. We elaborate regularization principles for reformulating the illposed...

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring ...