Some Analogs of the Generalized Principal Axis Transformation

It is known that two normal matrices can be diagonalized by the same unitary transformation if and only if they commute ; this theorem is ordinarily stated for hermitian matrices. Some generalizations of this theorem are known. According to a theorem due to Eckert and Young, if A and B are two rXs matrices, there are two unitary matrices U and V such that UAV=DX and UBV=D2, D\ and D2 diagonal matrices with real elements, if and only if AB and B A are hermitian. I t is also known that a set of normal matrices {Ai} is reducible to diagonal matrices under the same unitary similarity transformation, UAiU, if and only if AiA3= AjAi for all i and j . (More generally, it is true that a set of matrices {Ai} with elements in the complex field and simple elementary divisors is reducible to diagonal matrices under the same similarity transformation if and only if AiAj=AjAi for all i and j.) The following will be shown to hold: THEOREM. If {Ai} is an arbitrary set of nonzero rXs matrices, there are unitary matrices U and V of orders rXr and sXs, respectively, such that UAiV=Di, Di diagonal and real, if and only if AiAf—AjAf and AfAi = AfAjfor all i and j .