Permutation Matrix
Chapter 6: Geometric Transformation and Eigen Decomposition — Permutation
From the book
Chapter 6: Geometric Transformation and Eigen Decomposition. In the chapter mind map this icon labels Permutation. The discussion below is excerpted and lightly edited from § Permutation in Mathematics for AI and Machine Learning.
A permutation matrix $P_\sigma$ (the matrix chapter) reorders the coordinates of a vector. In 2D, swapping the two coordinates is the reflection $F(45^\circ)$ across the line $y = x$ (the book figure in 3D, cycling the three coordinates is a rotation by $120^\circ$ about the axis $(1,1,1)$:
It is orthogonal, $P_\sigma^\top = P_\sigma^{-1}$, so lengths and angles are preserved, and $\det(P_\sigma) = \mathrm{sgn}(\sigma)$ is $+1$ for an even permutation, like a rotation, and $-1$ for an odd one, like a reflection. In machine learning, relabeling the nodes of a graph or the elements of a set is a permutation, so models on graphs and sets are built to be permutation invariant: their output does not depend on the labeling.
What this drawing shows
What you see. Binary matrix that reorders rows or columns by permuting basis vectors.
In the mind map. Chapter 6 — Permutation. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 6: Geometric Transformation and Eigen Decomposition — see the mind-map node Permutation.
