Show that Q(0) . 0 for every quadratic form Q.
Since every quadratic form passes through the origin (exercise 3.93), a positive definite quadratic form has a unique minimum (at 0). Similarly a negative definite quadratic form has a unique maximum at 0. This hints at their practical importance in optimization. Consequently we need criteria to identify definite quadratic forms and matrices. Example 3.34 provides a complete characterization for 2 × 2 matrices. Conditions for definiteness in higher-dimensional spaces are analogous but more complicated (e.g., see Simon and Blume 1994, pp. 375±386; Sundaram 1996, pp. 50±55; Takayama 1985, pp. 121±123; Varian 1992, pp. 475±477.) Some partial criteria are given in the following exercises.