The aim of this paper is to describe the inversion of the sum n≥0 a n b n where a and b are non-commuting variables as a formal series in a and b. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals a Catalan number. We also study general solutions of similar… (More)
In this paper we proposes a generalized spectral theory for tensors. Our proposed factoriza-tion decomposes a symmetric tensor into a product of an orthogonal and a diagonal tensor. In the same time, our factorization offers an expansion of a tensor as a summation of lower rank tensors that are obtained through an outer product defined on matrices. Our… (More)
Given a matrix over a skew field fixing the column t (1,. .. , 1), we give formulas for a row vector fixed by this matrix. The same techniques are applied to give noncommutative extensions of probabilistic properties of codes.