Jordan–Chevalley Decomposition in Lie Algebras

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$ , then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$ , $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$ ) such that $A=S+N$ .

Dynamic Phasors Estimation Based on Taylor-Fourier Expansion and Gram Matrix Representation

The paper presents a new approach to estimation of the dynamic power phasors parameters. The observed system is modelled in algebra of matrices related to its Taylor-Fourier-trigonometric series representation. The proposed algorithm for determination of the unknown phasors parameters is based on the analytical expressions for elements of the Gram’s matrix associated with this system. The numerical complexity and algorithm time are determined and it is shown that new strategy for calculation of Gram’s matrix increases the accuracy of estimation, as well as the speed of the algorithm with respect to the classical way of introducing the Gram’s matrix. Several simulation examples of power system signals with a time-varying amplitude and phase parameters are given by which the robustness and accuracy of the new algorithm are confirmed.

Stable Subspaces of Positive Maps of Matrix Algebras

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

On the Lie structure of zero row sum and related matrices

We study the structure of zero row sum matrices as an algebra and as a Lie algebra in the context of groups preserving a given projection in the algebra of matrices. We find the structure of the Lie algebra of the group that fixes a given projection. Details for the zero row sum matrices are presented. In particular, we find the Levi decomposition and give an explicit unitary equivalence with the affine Lie algebra. An orthonormal basis for zero row sum matrices appears naturally.