Rotate Scale Rotate Decomposition

Why

Every matrix $A \in \R ^{m \times n}$ maps the unit ball in $\R ^n$ to an ellipsoid in $\R ^{m}$.

Definition

A rotate scale rotate decomposition (or rotate scale rotate factorization) of a matrix $A \in \R ^{m \times n}$ is an ordered triple $(U, S, V)$ where $U$ and $V$ are orthgonal and $S$ is diagonal decreasing ($S_{11} \geq S_{22} \geq \dots \geq S_{pp}$, where $p = \min\set{m, n}$) satisfying

\[ A = USV^\top . \]

Other (universal) terminology includes the singular value decomposition or SVD of $A$. We call diagonal elements of $S$ the singular values of $A$. We call the column vectors of $U$ the left singular vectors or output singular vectors. We call the column vectors of $V$ the right singular vectors or input singular vectors. We refer to them collectively as the singular vectors.

\[ Av_i = \sigma _i u_i. \]