Applications of matrix derivatives to optimization problems in statistical pattern recognition

A necessary condition for a real valued Frechet differentiable function of a vector variable have an extremum at a vector x sub 0 is that the Frechet derivative vanishes at x sub 0. A relationship between Frechet differentials and matrix derivatives was established that obtains a necessary condition on the matrix derivative at an extrema. These results are applied to various scalar functions of matrix variables which occur in statistical pattern recognition.