Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions H is the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy ...
Frobenius' Theorem states that the only finite-dimensional real division algebras are the algebra of real numbers R \mathbb R , the algebra of complex numbers C \mathbb C , and the algebra of quaternions H \mathbb H . We present a short proof which uses only standard undergraduate mathematics...
A classical theorem of Frobenius states that there are, to within isomorphism, only three finite-dimensional division algebras over ℝ, namely ℝ, ℂ, and the quaternion division algebra Q. Thus, if F = ℝ, the irreducible finite-dimensional representations T of A split into three classes...
1.This paper introduces an important Perron-Frobenius theorem in algebra,and discusses the first principal component served as the principle and condition of the system evaluation index.介绍了代数学中的一个重要定理(Perron-Frobenius定理),论述了第一主成分作为系统评估指数的原理和条件;对两类系统排序评估...
Theorem 2.1 implies that the finiteness of the p-Frobenius vec- tor, Fp(S), is independent of both the value p and the chosen graded mono- mial order. In simpler terms, if F1(S) exists, then all Fp(S) for p ≥ 1 also exist. Corollary 2.3. Let S = a1, . . . , ah ⊂ ...
2) Perron-Frobenius theorem Perron-Frobenius定理 1. This paper introduces an important Perron-Frobenius theorem in algebra,and discusses the first principal component served as the principle and condition of the system evaluation index. 介绍了代数学中的一个重要定理(Perron-Frobenius定理),论述了第一...
2) Perron-Frobenius theorem Perron-Frobenius定理 1. This paper introduces an important Perron-Frobenius theorem in algebra,and discusses the first principal component served as the principle and condition of the system evaluation index. 介绍了代数学中的一个重要定理(Perron-Frobenius定理),论述了第一...
In this article we give a new proof of the Fundamental Theorem of Algebra. Our proof is algebraic. We simplify the known proof of the Fundamental Theorem considering special case of polynomials of odd degree with real coefficients. This case allows us to apply the method of mathematical ...
Mazur''s theorem [19] states that every normed division algebra over the field of real numbers is isomorphic to either the field R of real numbers, the field C of complex numbers, or the non-commutative algebra Q of quaternions. Gelfand [15] proved that every normed division algebra over ...
Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for...