L. L. Thurstone, Am. J. Psychol. 38, 368–389 (1927)
F. Kottler, unpublished work.
J. P. Guilford, Psyclhomtetric Metllods (McGraw-Hill Book Company, Inc., New York, 1936), pp. 217–243.
It can be shown, however, that the Guilford-type solution is equivalent to the new method only if these apparent redundancies are not discarded. Their retention characterizes the "corrected form of the algorithm of J. P. Guilford" mentioned in the abstract. of this paper.
A convenient method for the inversion of integer-element matrices, using row transformations only, is given by J. Barkley Rosser in J. Research Natl. Bur. Standards 49, 349 (No. 5, November, 1952).
This departure, for derivation only, is temporary.
P. S. Dwyer, Ann. Math. Stat. (1)15, 82 (March, 1944).
An alternative procedure, which will not be further elaborated here, for avoiding the impasse resulting from singularity of A′A, where A′ is not augmented as described above, is to factor A into a product of two rank-(n-1) matrices B and M, where M consists of (n-1) rows, each specifying a differencing operation, and then to solve for the (n-1)-element vector represented by the matrix product Mr; this procedure involves the inversion of the (n-1) × (n-l) symmetric matrix B′B, which is generally nonsingular. An interesting feature of this solution for Mr, entirely equivalent to that for r, is its interpretation as the correct way to calculate, according to the least-squares principle, a set of (n-1) "basic differences," the objective of the Guilford "differencing" algorithm.
In this case, the new method degenerates into the Kottler "summation algorithm."
G. C. Higgins and R. N. Wolfe, J. Opt. Soc. Am. 45, 121 (1955).