A procedure is given for generating correlation matrices which can be used as population correlation matrices for sampling experiments. The algorithm specifies the eigenvalues and randomly selects a correlation matrix from the class of all correlation matrices which possess these same eigenvalues. It is possible to obtain a set of correlation matrices which are indexed by the degree of interdependence among the variables by parameterizing the eigenvalues with a single parameter. An example is the case in which the eigenvalues form a geometric progression. Examples are given and an application to the problem of stopping rules in stepwise regression is discussed. Other applications are also briefly discussed.