Orthomax Rotation Problem. a Differential Equation Approach1)

In the present paper the ORTHOMAX rotation problem is reconsidered. It is shown that its solution can be presented as a steepest ascent flow on the manifold of orthogonal matrices. A matrix formulation of the ORTHOMAX problem is given as an initial value problem for matrix differential equation of first order. The solution can be found by any available ODE numerical integrator. Thus the paper proposes a conver gent method for direct matrix solution of the ORTHOMAX problem. The well-known first order necessary condition for the VARIMAX maximizer is reestablished for the ORTHOMAX case without using Lagrange multipliers. Addition ally new second order optimality conditions are derived and as a consequence an explicit second order necessary condition for further classification of the ORTHOMAX maxim izer is obtained. 1 1. Introduction In factor analysis a decomposition of the sample n x n covariance/correlation matrix C is sought in the form : C=AA T+0, where A is n x p(n>>p) matrix of factor loadings and 0 is n x n diagonal positive definite matrix of unique variances. Such a representation is not unique. Let Q be any orthogonal p x p matrix and B=AQ. Then BB T =AQQTAT =AA', which means that the matrix of the factor loadings B yields the same covariance/correla tion matrix as A. This indeterminicity of the factor solution leads to the problem of finding the "best" transformation (rotation) Q of the factor loadings such that the rotated factors to have a structure, which to be as simple as possible for interpreta tion. Detailed consideration of the factor simplicity concept and many different methods for factor rotation can be found in Mulaik (1972). The VARIMAX rotation method (Kaiser, 1958; Mulaik, 1972) is the most popular one to achieve an orthogonal simple structure solution. The core of the method is to make most of the loadings in each factor of near zero magnitude and