SVD Mapping Geometry

Chapter 8: Singular Value Decomposition (SVD). In the chapter mind map this icon labels **Four Fundamental Subspaces Mapping. The discussion below is excerpted and lightly edited from § Four Fundamental Subspaces Mapping** in Mathematics for AI and Machine Learning.

Let $A \in \mathbb{R}^{M \times N}$ have rank $r$ and full SVD $A = U\Sigma V^\top$. Partitioning $U = [U_r \;\; \bar{U}_r]$ and $V = [V_r \;\; \bar{V}_r]$ at index $r$—with $U_r = [\mathbf u_0, \ldots, \mathbf u_{r-1}]$, $\bar{U}_r = [\mathbf u_r, \ldots, \mathbf u_{M-1}]$, $V_r = [\mathbf v_0, \ldots, \mathbf v_{r-1}]$, and $\bar{V}_r = [\mathbf v_r, \ldots, \mathbf v_{N-1}]$—yields an explicit orthonormal basis and orthogonal projector for each of the four fundamental subspaces (the matrix chapter):

1. Column space $C(A) \subseteq \mathbb{R}^M$: spanned by the columns of $U_r$, with orthogonal projector $P_{C(A)} = U_r U_r^\top$. 2. Row space $C(A^\top) \subseteq \mathbb{R}^N$: spanned by the columns of $V_r$, with orthogonal projector $P_{C(A^\top)} = V_r V_r^\top$. 3. Null space $N(A) \subseteq \mathbb{R}^N$: spanned by the columns of $\bar{V}_r$, with orthogonal projector $P_{N(A)} = \bar{V}_r \bar{V}_r^\top = I_N - V_r V_r^\top$. 4. Left null space $N(A^\top) \subseteq \mathbb{R}^M$: spanned by the columns of $\bar{U}_r$, with orthogonal projector $P_{N(A^\top)} = \bar{U}_r \bar{U}_r^\top = I_M - U_r U_r^\top$.

Singular vectors with $\sigma_i > 0$ span the two range spaces, while those with $\sigma_i = 0$ span the two null spaces.