I.—On a Problem concerning Matrices with Variable Diagonal Elements

Suppose that in a given non-negative definite symmetric matrix R = r ij of order n the diagonal entries are replaced by arbitrary quantities x 1 , x 2 ,…, x n so that the matrix assumes the form Matrices of this type are met with in the statistical technique known as Factorial Analysis (Thurstone, 1935; Thomson, 1939); there the nondiagonal elements r ij ( i≠j ) are the correlation coefficients of certain tests and are given by observation. The diagonal entries of the “correlational matrix,” which is always non-negative definite, are originally all equal to unity, but, on the hypothesis which underlies the process of Factorial Analysis, it is permissible to diminish the diagonal entries arbitrarily provided the modified matrix is still non-negative definite , it being assumed that the amount which has been deducted from the diagonal cells is due to the variance of the specific factors, the investigation of which is not the primary aim of the theory.