X-matrices

We evidence a family 𝒳 of square matrices over a field 𝕂, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that 𝒳 is a (in general non-commutative) subring of GL(n,𝕂). Moreover, we analyse the condition for a matrix A ∈𝒳 to be invertible in 𝒳. We also show that, if one adds a symmetry condition called here bi-symmetry, then the set 𝒳^b of bi-symmetric X-matrices is a commutative subring of 𝒳. We propose results for eigenvalue inclusion, showing that for X-matrices eigenvalues lie exactly on the boundary of Cassini ovals. It is shown that any monic polynomial on ℝ can be associated with a companion matrix in 𝒳.