During my course in linear algebra, the instructor stated that A cross B is the same as the "skew symmetric matrix" of A times B. So, first of all, can someone clarify or provide sources about skew symmetric matrices? Secondly, I can't really comprehend the idea of how a single column vector crossed with another could be represented by a matrix.
Anyhow, thanks in advance!
Best Answer
Imagine a column vector ${\bf A} = (A_1, A_2, A_3)$ and define the matrix
$$ A_\times = \left(\begin{array}{ccc} 0 & -A_3 & A_2 \\ A_3 & 0 & -A_1 \\ -A_2 & A_1 & 0 \end{array}\right) $$
Note that if ${\bf B}$ is another column vector, then
$$ A_\times {\bf B} = {\bf A}\times {\bf B} $$
Moreover
$$ {\rm Transpose}(A_\times) = -A_\times $$