Hidden relations for the Smith invariants of a matrix sum
(Homogeneous) Markov bridges are (time homogeneous) Markov chains which begin at a given point and end at a given point. The price to pay for preserving the homogeneity is to work with processes with a random life-span. Bridges are studied both for themselves and for their use in describing the transformations of Markov chains: restriction on a random interval, time reversal, time change, various conditionings comprising the confinement in some part of the state space. These bridges lead us to look...
Let be the multiplicative semigroup of all complex matrices, and let and be the –degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from to when or , and thereby determine multiplicative homomorphisms from to when or . This generalize Hochwald’s result in [Lin. Alg. Appl. 212/213:339-351(1994)]: if is a spectrum–preserving multiplicative homomorphism, then there exists a matrix in such that for...
Necessary and sufficient conditions are presented for the commutativity equalities , , , and so on to hold by using rank equalities of matrices. Some related topics are also examined.
After an overview of Hurwitz pairs we are showing how to actually construct them and discussing whether, for a given representation, all Hurwitz pairs of the same type are equivalent. Finally modules over a Clifford algebra are considered with compatible inner products; the results being then aplied to Hurwitz pairs.
The article is devoted to a generalization of Clifford and Grassmann algebras for the case of vector spaces over the field of complex numbers. The geometric interpretation of such generalizations are presented. Multieuclidean geometry is considered as well as the importance of it in physics.