Raising the bar

Permutation matrices

Definition and example

Let $latex e_i=(0,...,0,1,0,...0)$ denote the $latex i$th unit row-vector (the $latex i$th component is 1 and the others are zeros).

Definition 1. A permutation matrix $latex P$ is defined by two conditions: a) all of its rows are unit row-vectors and b) no two rows are equal. We use the notation

$latex P_{j_1,...,j_n}=\left(\begin{array}{c}e_{j_1} \\... \\e_{j_n}\end{array}\right)$

where all rows are different.

Example. In the following matrix we change places of rows in such a way as to obtain the identity in the end. Each transformation changes the sign of the matrix by Property VI.

$latex P_{2,4,1,3}=\left(\begin{array}{c}e_2 \\e_4 \\e_1 \\e_3\end{array}\right) =\left(\begin{array}{cccc}0&1&0&0 \\0&0&0&1 \\1&0&0&0 \\0&0&1&0\end{array}\right)\overset{\text{changing 1st and 3rd rows}}{\rightarrow }\left(\begin{array}{cccc}1 &0&0&0 \\0&0&0&1 \\0&1&0&0 \\0&0&1&0\end{array}\right) $

$latex \overset{\text{changing 3rd and 4th rows}}{\rightarrow }\left(\begin{array}{cccc}1&0&0&0 \\0&0&0&1 \\0&0&1&0 \\0&1&0&0
\end{array}\right)\overset{\text{changing 2nd and 4th rows}}{\rightarrow }\left(\begin{array}{cccc}1&0&0&0 \\
0&1&0&0 \\0&0&1&0 \\0&0&0&1\end{array}\right).$

Since there are three changes and the final matrix is identity, by Axiom 3 $latex \det P_{2,4,1,3}=-1$.

It is possible to arrive to the identity matrix in more than one way but the result will be the same because the number $latex \det P_{2,4,1,3}$ is fixed. Calculating determinants of permutation matrices in this way allows us to avoid studying the theory of permutations.

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