I have a problem. From wikipedia http://en.wikipedia.org/wiki/Positive-definite_matrix any function can be written as $$z^TMz$$ where z is a column vector and M is a symmetric real matrix. However this quadratic function is strictly convex only when M is symmetric positive definite. Why ?, I thought any quadatic function should be convex ? doesn't $$z^TMz$$ >0 shows only that the range of this function is greater than zero? 1. Why isn't any symmetric matrix M(which represents a quadratic function) convex ?
- Why is it only the case when $$z^TMz$$ denotes convex ?