JacobisTheorem 2003-04-05 11:52:46 2006-09-13 21:59:28 Theorem skew-symmetric matrix % 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 {\bf Jacobi's Theorem} Any skew-symmetric matrix of odd order has determinant equal to $0$. {\bf Proof.} Suppose $A$ is an $n\times n$ square matrix. For the determinant, we then have $\det A = \det A^T$, and $\det (-A) = (-1)^n \det A$. Thus, since $n$ is odd, and $A^T=-A$, we have $\det A = -\det A$, and the theorem follows. $\Box$ \subsubsection{Remarks} \begin{enumerate} \item According to \cite{eves}, this theorem was given by Carl Gustav Jacob Jacobi (1804-1851) \cite{jacobi} in 1827. \item The $2\times 2$ matrix $\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$ shows that Jacobi's theorem does not hold for $2\times 2$ matrices. The determinant of the $2n\times 2n$ block matrix with these $2\times 2$ matrices on the diagonal equals $(-1)^n$. Thus Jacobi's theorem does not hold for matrices of even order. \item For $n=3$, any antisymmetric matrix $A$ can be written as $$ A = \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix} $$ for some real $v_1,v_2,v_3$, which can be written as a vector $v=(v_1,v_2,v_3)$. Then $A$ is the matrix representing the mapping $u\mapsto v\times u$, that is, the cross product with respect to $v$. Since $Av=v\times v=0$, we have $\det A=0$. \end{enumerate} \begin{thebibliography}{9} \bibitem {eves} H. Eves, \emph{Elementary Matrix Theory}, Dover publications, 1980. \bibitem{jacobi} The MacTutor History of Mathematics archive, \PMlinkexternal{Carl Gustav Jacob Jacobi}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html} \end{thebibliography}