11
Feb 19

Permutation matrices




Permutation matrices

Definition and example

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

Definition 1. A permutation matrix 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

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.

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)

\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 \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 \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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