Orthogonal and Scaling Transformations of Quadratic Functions with Some Applications

Abstract

In this paper we present a non-singular transformation that can reduce a given quadratic function defined on Rⁿ to another simpler quadratic function and study the impact of the transformation in relation to the problem of minimization of the function. In particular, we construct a non-singular transformation that can reduce a quadratic function whose level surfaces are ellipsoids to another quadratic function whose level surfaces are spherical while preserving the convexity/concavity property of the given function. The relation between a minimizing point of the given function and that of the new simpler function obtained under the transformation is also described.