In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector (span of the generators) on homogeneous i.e. sums of decomposable (Graßmann), multi-vectors and later extended by bilinearity. The Hestenesian ’dot’ product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need...

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we...

In this paper we generalize the known De Smet, Dillen, Verstraelen and Vrancken (DDVV)-type inequalities for real (skew-)symmetric complex (skew-)Hermitian matrices to arbitrary real, quaternionic matrices. Inspired by Erdős-Mordell inequality, establish DDVV-type in subspaces spanned a Clifford system or algebra. We also Böttcher-Wenzel inequality

Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann algebras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean space. É. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do ...

Polyvector-valued gauge field theories in noncommutative Clifford spaces are presented. The noncommutative binary star products are associative and require the use of the Baker-Campbell-Hausdorff formula. An important relationship among the n-ary commutators of noncommuting spacetime coordinates [X, X, ......, X] and the poly-vector valued coordinates X in noncommutative Clifford spaces is expl...

It is known that the Williamson construction for Hadamard matrices can be generalized to constructions using sums of tensor products. This paper describes a specific construction using real monomial representations of Clifford algebras, and its connection with graphs of amicability and anti-amicability. It is proven that this construction works for all such representations where the order of th...

A class of matrix valued functions defined by singular values of nonsymmetric matrices are shown to have many properties analogous to matrix valued functions defined by eigenvalues of symmetric matrices. In particular, the (smoothed) matrix valued Fischer-Burmeister function is proved to be strongly semismooth everywhere. This result is also used to show the strong semismoothness of the (smooth...

Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is e...