The technique of matrix rank analysis has many applications in biochemistry and molecular biology. It can be applied to determine the number of components in a mixture of absorbing species,'.z to analyze optical rotatory dispersion data of the RNA of tobacco mosaic virus,$ and to determine the number of components in fluorescent spectros~opy.~.~ Furthermore, it can be applied to the kinetics of the oxidation of hemoglobin derivatives: to gel-filtration experiments,? to the determination of components from a diffusion experiment in the ultracentrifuge: and to work on membrane^.^ Fundamentally, a matrix array of numbers is the result of any of the above experiments. In determining the rank of the matrix, the number of linearly independent components will be identified. One cannot determine the rank by simply finding the smallest non-zero minor, since the matrix elements, being experimental measurements, are subject to error. Because of experimental error, it is probable that the entire data matrix under consideration is non-singular.
To determine the rank of the matrix, Wallace' first suggested that the values and probable error of the determinants be computed and compared. This procedure was felt to be too tedious. Later Wallace and Katz2 suggested a procedure whereby the original data matrix and a comparison error matrix are reduced simultaneously. The rank of the data matrix was then decided by comparing the elements of the reduced original matrix and the reduced error matrix.
Both of the above criteria depend upon the computation of probable error using error propagation equations.9 These equations generally assume that the errors are both independent and small in magnitude. The independence of the error is probably a good assumption, considering the large number of observations.'0 On the other hand, the assumption of the magnitude of error may or may not be correct.
An alternative method t o matrix rank analysis is factor analysis, which provides the following information and improvements:
- The number of components in the system is given with a certain probability level.
For any given sample we can estimate how much of its response is due to a given For example, if the system was found to be made of two factors (or two com-