The Symmetrical Idempotent Matrix in Factor Analysis

It is of interest to note the occurrence in facI tor analysis of the symmetrical idempotent ma trix and to examine the role it may play as a mul tiplier of other matrices. For example, Holzin ger and Harman's method of relating two differ ent factorial solutions of the same correlation ma trix1 may be formulated, alternatively, in terms of the symmetrical idempotent matrix as a multi plication unit for certain factor matrices. Tuck er's semi-analytic method of rotating to a simple structure, 2 which utilizes the same equation as that given by Holzinger and Harman, also maybe formulated in these terms, providing the effect of using the symmetrical idempotent as a multi plier of matrices for which it is not a unit for multiplication is noted. The Holzinger-Harman treatment is concerned with the problem of solv ing for a transformation matrix that rotates one given factorial solution into a second, and given, one. Tucker's problem differs, in that the sec ond solution is not given but is constructed from an initial arbitrary matrix through a series of ap proximations; he therefore introduces the require ment of a best fit in the least squares sense and shows that the type of matric equation employed meets this requirement. Somewhat earlier, Horst had proposed a rotation method, using similar principles, that involved the actual calculation of a symmetrical idempotent matrix and the use of its entries as a guide to the rotation process. 3 These examples suggest that a review of the idem potent as a particular type of matric element may aid in clarifying certain algebraic principles com mon to these several papers and in outlining cer| tain applications of these principles in factor an alysis. As a preliminary step, consider the equation: