PositiveDefiniteForm 2002-02-22 03:17:55 2004-09-20 18:50:55 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A bilinear form $B$ on a real or complex vector space $V$ is {\em positive definite} if $B(x,x) > 0$ for all nonzero vectors $x \in V$. On the other hand, if $B(x,x) < 0$ for all nonzero vectors $x \in V$, then we say $B$ is {\em negative definite}. If $B(x,x) \ge 0$ for all vectors $x \in V$, then we say $B$ is {\em nonnegative definite}. Likewise, if $B(x,x) \le 0$ for all vectors $x \in V$, then we say $B$ is {\em nonpositive definite}. A form which is neither positive definite nor negative definite is called {\em indefinite}.