Examples are presented in which it is either desirable or necessary to transform two sets of orthogonal axes to simple structure positions by means of the same transformation matrix. A solution is then outlined which represents a two-matrix extension of the general “orthomax” orthogonal rotation criterion. In certain circumstances, oblique two-matrix solutions are possible using the procedure outlined and the Harris-Kaiser [1964] logic. Finally, an illustrative example is presented in which the preceding technique is applied in the context of an inter-battery factor analysis.