On the Jordan form of a family of linear mappings (Q1355216)

Let \(A\) and \(B\), respectively, be \(m\times m\) and \(n\times n\) matrices over a field \(F\), and let \(g(x,y)=\sum a_{rs}x^ry^s\) be a polynomial over \(F\). The author considers the problem of describing the Jordan canonical form for \(T:=\sum a_{rs}(A^r\otimes B^s)\) in terms of the Jordan canonical forms for \(A\) and \(B\). The cases \(g(x,y)=x+y\) and \(g(x,y)=xy\) in which \(T\) corresponds to the linear mappings \(X\mapsto A^TX+XB\) and \(X\mapsto A^TXB\) have been extensively studied. The author obtains a complete description of the Jordan form for \(T\) in the general case provided the partial derivatives \(\partial g/\partial x\) and \(\partial g/\partial y\) do not vanish at any pair \((\lambda,\mu)\) of eigenvalues for \(A\) and \(B\). He illustrates his results by a detailed discussion of the \(2\times 2\) and \(3\times 3\) cases.