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The Jordan Form Problem for C=AB: the Balanced, Diagonalizable Case

We consider a key case in the large problem of the possible Jordan canonical forms of A,B,C \in M_n(F) when C=AB. If A \in M_{2k}(F) (respectively B,C \in M_{2k}(F)) is diagonalizabel with two distinct eigenvalues a_1, a_2 (respectively b_1, b_2 and c_1, c_2), each with multiplicity k, and when C=AB, all possibilities for a_1, a_2, b_1, b_2, c_1, c_2 are characterized. The possibilities are much more restrictive than the obvious determinant condition: (a_1a_2b_1b_2)^k= (c_1c_2)^k allows. This is then used to settle the general, two eigenvalue per matrix, diagonalizable case of the Jordan form problem for C=AB.

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