Permutation Matrix - Properties

Properties

Given two permutations π and σ of m elements and the corresponding permutation matrices Pπ and Pσ

This somewhat unfortunate rule is a consequence of the definitions of multiplication of permutations (composition of bijections) and of matrices, and of the choice of using the vectors as rows of the permutation matrix; if one had used columns instead then the product above would have been equal to with the permutations in their original order.

As permutation matrices are orthogonal matrices (i.e., ), the inverse matrix exists and can be written as

Multiplying times a column vector g will permute the rows of the vector:

P_\pi \mathbf{g} = \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \vdots \\ \mathbf{e}_{\pi(n)} \end{bmatrix} \begin{bmatrix} g_1 \\ g_2 \\ \vdots \\ g_n \end{bmatrix} = \begin{bmatrix} g_{\pi(1)} \\ g_{\pi(2)} \\ \vdots \\ g_{\pi(n)} \end{bmatrix}.

Now applying after applying gives the same result as applying directly, in accordance with the above multiplication rule: call, in other words

for all i, then

P_\sigma(P_\pi(\mathbf{g})) = P_\sigma(\mathbf{g}') = \begin{bmatrix} g'_{\sigma(1)} \\ g'_{\sigma(2)} \\ \vdots \\ g'_{\sigma(n)} \end{bmatrix} = \begin{bmatrix} g_{\pi(\sigma(1))} \\ g_{\pi(\sigma(2))} \\ \vdots \\ g_{\pi(\sigma(n))} \end{bmatrix}.

Multiplying a row vector h times will permute the columns of the vector by the inverse of :

\mathbf{h}P_\pi = \begin{bmatrix} h_1 \; h_2 \; \dots \; h_n \end{bmatrix} \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \vdots \\ \mathbf{e}_{\pi(n)} \end{bmatrix} = \begin{bmatrix} h_{\pi^{-1}(1)} \; h_{\pi^{-1}(2)} \; \dots \; h_{\pi^{-1}(n)} \end{bmatrix}

Again it can be checked that .

Read more about this topic:  Permutation Matrix

Famous quotes containing the word properties:

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)