We consider a strongly regular graph, G, with adjacency matrix A, and associate a three dimensional Euclidean Jordan algebra to A. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of the Euclidean Jordan algebra associated to A, we establish new admissibility conditions for the existence of strongly regular grap...

Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If is generated as by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ arbitrary vector space $F$, the condition that $\Phi(x^2,x)=0 $ all $x\in A$ implies $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ $x,y,z\in A$. This applicable to question of whether zero Lie product determined and also used in pro...

Let A be a matrix and λ0 be one of its eigenvalues having g elementary Jordan blocks in the Jordan canonical form of A. We show that for most matrices B satisfying rank (B) ≤ g, the Jordan blocks of A+B with eigenvalue λ0 are just the g− rank (B) smallest Jordan blocks of A with eigenvalue λ0. The set of matrices for which this behavior does not happen is explicitly characterized through a scal...

A square complex matrix A is eventually nonnegative if there exists a positive integer k0 such that for all k ≥ k0, A ≥ 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jorda...