SemiDefinite What?

This page contains definitions for SemiDefinite things like matrices, programs, etc.

== SemiDefinite Matrices ==

A positive SemiDefinite matrix is a HermitianMatrix all of whose eigenvalues are nonnegative. Thus any symmetric matrix that has a 0 on the diagonal is a SemiDefinite matrix.

SemiDefinite Programming

\usepackage{amsmath}%
\setcounter{MaxMatrixCols}{30}%
\usepackage{amsfonts}%
\usepackage{amssymb}%
\usepackage{graphicx}
\usepackage{geometry}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.5in}

%%end-prologue%%
Suppose you have a loop in a program where the values of the variable values at the start of the loop are denoted $x_0 \in \mathbb{R}^n$ and the values at the end of the loop are denoted $x$ where $n$ is the number of variables.

Let $M(x)=M_0 + \sum\limits^m_{k=1} x_k.M_k