# Definition of permutation matrix

Put content here**Definition:** A **permutation matrix** is a square binary matrix obtained by permuting the rows (or columns) of the identity matrix. Each row and each column contains exactly one 1, with all other entries being 0.
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**Example:** Permuting rows of $I_3$:
$$P = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$
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**Properties:**
- $P^{-1} = P^T$ (permutation matrices are orthogonal)
- $\det(P) = \pm 1$ (sign depends on permutation parity)
- Multiplying $PA$ permutes the rows of $A$
- Multiplying $AP^T$ permutes the columns of $A$
- $P^k = I$ for some positive integer $k$
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Permutation matrices are used in **LU decomposition with partial pivoting** ($PA = LU$) and in representing permutations as linear transformations.

# Parents

* Particular types of matrices