A matrix generalization of Euler identity e jϕ = cosϕ + j sinϕ
نویسنده
چکیده
In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential representation for matrix real and imaginary unit, and for the first Pauli matrix.
منابع مشابه
The Perles-Shephard identity for non-convex polytopes
Using the theory of valuations, we establish a generalization of an identity of Perles-Shephard for non-convex polytopes. By considering spherical valuations, we obtain the Gram-Euler, Descartes and Euler-Poincar e theorems for non-convex polytopes.
متن کاملOn strongly J-clean rings associated with polynomial identity g(x) = 0
In this paper, we introduce the new notion of strongly J-clean rings associated with polynomial identity g(x) = 0, as a generalization of strongly J-clean rings. We denote strongly J-clean rings associated with polynomial identity g(x) = 0 by strongly g(x)-J-clean rings. Next, we investigate some properties of strongly g(x)-J-clean.
متن کاملThe r-matrix structure of the Euler-Calogero-Moser model
We construct the r-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the r-matrix for the O(N) Euler-Calogero-Moser model and for the standard AN Calogero-Moser model. PAR LPTHE 93-55 L.P.T.H.E. Université Paris VI (CNRS UA 280), Box 126, Tour 16, 1 étage, 4 place Jussieu, F-75252 PARIS CEDEX 05
متن کاملCombinatorial Aspectsof Multiple
Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shu e product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with c...
متن کاملBernoulli Number Identities from Quantum Field Theory
We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli ...
متن کامل