392 CHAPTER 9 then dividing by the product e gives (ln e)/e (resp. (ln )/). Now we use the fact that taking the natural logarithm of two numbers does not change their ordering, as the function ln is monotonically increasing. This shows that to prove that e > e it suffices to prove that (ln e)/e > (ln )/. This formulation suggests considering the function (ln x)/x (Fig. 9.1): 1 2 e ln x . x Figure 9.1 The graph of the function 1. f (x) = (ln x)/x max, x > 0. Existence of a solution x follows from the Weierstrass theorem, as f (x) < 0 x < 1, f (1) = 0, f (x) > 0 x > 1, and lim x+ f (x) = 0. 2. Fermat: f (x) = 0 (1 - ln x)/x 2 = 0. 3. x = e, f (e) = 1/e. 4. x = e, and so f (e) > f () e > e . 9.2 OPTIMIZATION APPROACH TO MATRICES In this section, we illustrate by three more examples a nonstandard way of proving basic results of matrix theory (see section 9.5 for two other examples). The traditional way to prove results on matrices is by means of the Gaussian elimination algorithm. In view of this, you might find it refreshing to see an alternative way: by optimiza- tion methods. Note that most results of matrix theory seem to have nothing to do with optimization.