3. Some Matrix Problems
Society for Industrial and Applied Mathematics eBooks · 1994
3.1 Minimizing Condition Number by Scaling The condition number of a matrix , with , is the ratio of its largest and smallest singular values, i.e., κ (M) ≜ λmax ( MT M) λmin ( MT M) 1/2 for M full-rank, and otherwise. For square invertible matrices this reduces to . We consider the problem: minimizeκ (LMR) subject toL∈ Rp×p ,diagonal and nonsingularR∈ Rq×q ,diagonal and nonsingular 3.1 where L and R are the optimization variables, and the matrix is given. We will show that this problem can be transformed into a GEVP. We assume without loss of generality that and M is full-rank.