A general solution to the maximization of the multidimensional generalized Rayleigh quotient used in linear discriminant analysis for signal classification

A more general solution and a new didactic demonstration of the maximization of the multidimensional case of the generalized Rayleigh quotient are described. This solution is not only the well-known eigenvectors solution widely available in the literature but also a general transformation that is not necessarily orthogonal. The demonstration uses only basic linear algebra and simple Lagrangian maximization to find the transformation matrix that maximizes the multidimensional generalized Rayleigh quotient for linear discriminant analysis, widely used in signal classification applications.