Relations among Factors of Raw, Deviation, and Double-Centered Score Matrices
THE PURPOSE of this paper is to develop relations among factors of raw scores, deviation scores, and double-centered scores. The treat ment is abstract and is merely an application of the algebra to the specialized matrices in which the interest of the factor analyst centers. Equiv alence of matrices, either square or rectangular, and similarity of square matrices are general notions that underlie the discussion. In addition, use is made of a specialized case of equivalence of matrices that occurs if the symmetric matrices generated by these matrices are elements of the same multiplicative group. Finally, a "loose" relation among matrices that generate symmetric matrices of different ranks belonging to the same sub-ring, or sub-algebra, of the total matric ring is employed to describe some of the results. It will be assumed that only real score matrices are employed and consequently that the symmetric ma trices they generate are not only real but also Gramian, that is, have no negative principal min ors. This is merely an arbitrary choice of fields, made because scores ordinarily are taken to be real numbers. The plan of the paper is first to describe a no tation that permits writing deviation scores as the matric product of raw scores and an idempo tent matrix. This notation should have a number of uses, a few of which are illustrated in this pa per. Next, the resolution of conventional devia tion scores into the product of factors and factor scores is summarized. This principle is, of course, well known; as it is summarized here, explicit use is made of the notion of groups o f singular matrices for which a symmetric idem potent matrix is a unit for multiplication. With the method established, it is possible to write equations relating the factors of raw scores to the factors of deviation scores, and equations re lating the factors of the two types of deviation scores. These results give a precise statement of Burt's reciprocity principle. Finally, double centered matrices are considered. Apparently two types of double-centered matrices may be identified; the relation of factors of such matrices to the factors of raw or deviation scores is out lined. It should be noted that an important restriction is made in this study. Throughout, it is assumed that the total variation exhibited in a set of data is to be analyzed in terms of common factors. This is a choice that the factor analyst may make. The results given here outline possibilities for relating solutions if he makes such a choice.