Offers a method to simplify J x J x 2 arrays and shows that the transformation that simplifies an I x J x K array can also be used to simplify the complementary arrays of three different orders. Discusses the maximal simplicity for arrays. (SLD)

Provides a method for generating the class of all solutions (or at least a subset of that class) given a CANDECOMP/PARAFAC (CP) solution that satisfies certain conditions. Shows mathematically that the condition defined by J. Kruskal is necessary and sufficient when the rank of the solution is three, and it may hold for higher ranks. (SLD)

Procedures for oblique rotation of factors or principal components typically focus on rotating the pattern matrix so that it becomes optimally simple. How the Harris and Kaiser independent cluster (1964) rotation can be modified to obtain a simple weights matrix rather than a simple pattern is described and illustrated. (SLD)

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has…

Developed a closed form expression for the asymptotic bias of the explained common variance, or the unexplained common variance under assumptions of multivariate normality in minimum rank factor analysis. Findings from existing data sets show that the presented asymptotic statistical inference is based on a recently developed perturbation theory…

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution that need not be centered row-wise. The results bear directly on several types of preprocessing methods in Parafac/Candecomp. (Author/TJH)

A summary and a unified treatment of fully general computational solutions for two criteria for transforming two or more matrices to maximal agreement are provided. The two criteria--Maxdiff and Maxbet--have applications in the rotation of factor loading or configuration matrices to maximal agreement and the canonical correlation problem. (SLD)

The phenomenon of 2 x 2 x 2 arrays having nonmaximal rank with positive probability, pointed out by J. Kruskal (1989), is generalized to 2 x "n" x "n" arrays. It is concluded that a pair of asymmetric square matrices can be diagonalized simultaneously with positive probability. (SLD)

Discusses sampling bias problems in the use of the greatest lower bound (g.l.b.) to reliability and offers explicit expressions for the second order derivatives. This yields closed form expression for the asymptotic bias of both the g.l.b. and its numerator. Illustrates the approach through a numeric example. (SLD)

A procedure is described for minimizing a class of matrix trace functions, which is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. Several trial analyses demonstrate that the revised procedure is more efficient than the earlier majorization-based procedure. (SLD)

A globally optimal solution is presented for a class of functions composed of a linear regression function and a penalty function for the sums of squared regression weights. A completing-the-squares approach is used, rather than calculus, because it yields global minimality easily in two of three cases examined. (SLD)

Arguments by J. Levin (1988) challenging the convergence properties of the Harman and Jones (1966) method of Minres factor analysis are shown to be invalid. Claims about the invalidity of a rank-one version of the Harman and Jones method are also refuted. (TJH)

An algorithm, based on a solution for C. I. Mosier's oblique Procrustes rotation problem, is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. Results are of interest for missing value and tetrachoric correlation, indefinite matrix correlation, and constrained…

Some uniqueness properties are presented for the PARAFAC2 model for covariance matrices, focusing on uniqueness in the rank two case of PARAFAC2. PARAFAC2 is shown to be usually unique with four matrices, but not unique with three unless a certain additional assumption is introduced. (SLD)

R. A. Bailey and J. C. Gower explored approximating a symmetric matrix "B" by another, "C," in the least squares sense when the squared discrepancies for diagonal elements receive specific nonunit weights. A solution is proposed where "C" is constrained to be positive semidefinite and of a fixed rank. (SLD)