Chapter 8: Singular Value Decomposition (SVD). In the chapter mind map this icon labels **Trans Geometry: Unit Sphere-to-Ellipse Mapping. The discussion below is excerpted and lightly edited from § Unit Sphere-to-Ellipse Mapping** in Mathematics for AI and Machine Learning. Related material also appears in Chapter 13 (Trans: Jacobian change of variables).

Let $A \in \mathbb{R}^{M \times N}$ be with rank $r$ and the following reduced SVD:

Then for any $\mathbf x \in \mathbb{R}^{N \times 1}$, $A\mathbf x$ can be interpreted as a linear transformation mapping $\mathbf x \in \mathrm{span}(V_r)$ in $\mathbb{R}^N$ to $\mathrm{span}(U_r)$ in $\mathbb{R}^M$.

Specifically, which decomposes into: (1) projection $V_r^\top \mathbf x$ onto $\mathrm{span}(V_r)$ (the row space of $A$), (2) scaling by singular values $\Sigma_r$, and (3) mapping to $\mathrm{span}(U_r)$ (the column space of $A$) via $U_r$.

The transformation $A$ acts as a rank-$r$ linear map, compressing $\mathbb{R}^N$ to an $r$-dimensional intermediate space, then expanding to $\mathbb{R}^M$.