is an orthogonal matrix and diag( is the diagonal matrix of eigenvalues of the eigenvalues are strictly positivesince is positive definite denoting the columns of by (qi )we have li > aqi tr qi aq> ( from the properties of determinantswe have :ln |aln | qln(| | | |pp ln |di= ln li we can thus write ln |ax ln li li ln li li li li = = = where the second equation follows from the chain rule applied to the function composition li from ( and example we have li / qi > it follows that qq - qa- = li objective function optimization theory optimization is concerned with finding minimal or maximal solutions of real-valued objective function in some set xmin (xor xx local minimizer global minimizer max (xxx ( since any maximization problem can easily be converted into minimization problem via the equivalence max (xmin ( )we focus only on minimization problems we use the following terminology local minimizer of (xis an element xx such that ( (xfor all in some neighborhood of xif ( (xfor all xthen xis called global minimizer or global solution the set of global minimizers is denoted by argmin (xxx local/global minimum the function value (xcorresponding to local/global minimizer xis referred to as the local/global minimum of (xoptimization problems may be classified by the set and the objective function if is countablethe optimization problem is called discrete or combinatorial if instead is nondenumerable set such as rn and takes values in nondenumerable setthen the problem is said to be continuous optimization problems that are neither discrete nor continuous are said to be mixed the search set is often defined by means of constraints standard setting for constrained optimization (minimizationis the followingmin (xxy subject tohi ( gi ( mi (