The jordan form problem for C=AB : the balanced, diagonalizable case

Abstract

We consider a key case in the fundamental and substantial prob- lem of the possible Jordan canonical forms of A;B;C \in Mn(F) when C = AB. If A \in M2k(F) (respectively B;C \in M2k(F) ) is diagonalizable with two distinct eigenvalues a1; a2 (respectively b1; b2, and c1; c2), each with multiplicity k, and when C = AB, all possibilities for a1; a2; b1; b2; c1; c2 are characterized. The possibilities are much more restrictive than the ob- vious determinant condition: (a1a2b1b2)k = (c1c2)k allows. This is then used to settle the general, two eigenvalue per matrix, diagonalizable case of the Jordan form problem for C = AB.