Abstract :
Let A and B be square matrices over a field F having their eigenvalues λ and μ in F, and let g(x, y) = ∑r, sarsxrys be a polynomial over F. Assuming the Jordan forms of A and B to be known, the Jordan form of ∑r,sarsAr circle times operator Bs is determined when the partial derivatives ∂g/∂x and ∂g/∂y are nonzero at (λ, μ), thus generalizing the classical cases g(x, y) = x + y and g(x, y) = xy. The particular cases g(x, y) = xk + yl and g(x, y) = xkyl are also generalized by using properties of the partial Hasse derivatives of g at (λ, μ). The case where A and B are 2 × 2 or 3 × 3 matrices, g being arbitrary, is discussed exhaustively.
