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Version 6 |
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{\bf Jacobi's Theorem} If $A$ is a skew-symmetric matrix of odd order, then $\det A = 0$.
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{\bf Jacobi's Theorem} If $A$ is a skew-symmetric matrix of odd dimension, then $\det A = 0$.
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| {\bf Proof.} Suppose $A$ is an $n\times n$ square matrix. |
{\bf Proof.} Suppose $A$ is an $n\times n$ square matrix. |
| For the determinant, we then have $\det A = \det A^T$, and |
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 |
$\det (-A) = (-1)^n \det A$. Thus, since $n$ is odd, and $A^T=-A$, we have |
| $\det A = -\det A$, and the theorem follows. $\Box$ |
$\det A = -\det A$, and the theorem follows. $\Box$ |
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| \subsubsection{Remarks} |
\subsubsection{Remarks} |
| \begin{enumerate} |
\begin{enumerate} |
| \item According to \cite{eves}, this theorem was given by |
\item According to \cite{eves}, this theorem was given by |
| Carl Gustav Jacob Jacobi (1804-1851) \cite{jacobi} in 1827. |
Carl Gustav Jacob Jacobi (1804-1851) \cite{jacobi} in 1827. |
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| \item The $2\times 2$ matrix $\left( \begin{array}{cc} |
\item The $2\times 2$ matrix $\left( \begin{array}{cc} |
| 0 & 1 \\ |
0 & 1 \\ |
| -1 & 0 |
-1 & 0 |
| \end{array} \right)$ shows that Jacobi's theorem does not hold for $2\times 2$ |
\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 |
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 |
these $2\times 2$ matrices on the diagonal equals $(-1)^n$. Thus Jacobi's theorem |
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does not hold for matrices of even order.
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does not hold for matrices of even dimension.
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| \item For $n=3$, any antisymmetric matrix $A$ can be written |
\item For $n=3$, any antisymmetric matrix $A$ can be written |
| as |
as |
| $$ A = |
$$ A = |
| \begin{pmatrix} |
\begin{pmatrix} |
| 0 & -v_3 & v_2 \\ |
0 & -v_3 & v_2 \\ |
| v_3 & 0 & -v_1 \\ |
v_3 & 0 & -v_1 \\ |
| -v_2 & v_1 & 0 |
-v_2 & v_1 & 0 |
| \end{pmatrix} |
\end{pmatrix} |
| $$ |
$$ |
| for some real $v_1,v_2,v_3$, which can be written as a |
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 |
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 |
mapping $u\mapsto v\times u$, that is, the cross product with |
| respect to $v$. Since $Av=v\times v=0$, we have $\det A=0$. |
respect to $v$. Since $Av=v\times v=0$, we have $\det A=0$. |
| \end{enumerate} |
\end{enumerate} |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {eves} H. Eves, |
\bibitem {eves} H. Eves, |
| \emph{Elementary Matrix Theory}, |
\emph{Elementary Matrix Theory}, |
| Dover publications, 1980. |
Dover publications, 1980. |
| \bibitem{jacobi} |
\bibitem{jacobi} |
| The MacTutor History of Mathematics archive, |
The MacTutor History of Mathematics archive, |
| \PMlinkexternal{Carl Gustav Jacob Jacobi}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html} |
\PMlinkexternal{Carl Gustav Jacob Jacobi}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html} |
| \end{thebibliography} |
\end{thebibliography} |
