Generalized Inverses of Matrix Products: From Fundamental Subspaces to Randomized Decompositions

Abstract

We investigate the Moore-Penrose pseudoinverse and generalized inverse of a matrix product $A=CR$ to establish a unifying framework for generalized and randomized matrix inverses. This analysis is rooted in first principles, focusing on the geometry of the four fundamental subspaces. We examine: (1) the reverse order law, $A^+ = R^+C^+$, which holds when $C$ has independent columns and $R$ has independent rows, (2) the universally correct formula, $A^+ = (C^+CR)^+(CRR^+)^+$, providing a geometric interpretation of the mappings between the involved subspaces, (3) a new generalized randomized formula, $A^+_p = (P^TA)^+P^TAQ(AQ)^+$, which gives $A^+_p = A^+$ if and only if the sketching matrices $P$ and $Q$ preserve the rank of $A$, i.e., $\mathrm{rank}(P^TA) = \mathrm{rank}(AQ) = \mathrm{rank}(A)$. The framework is extended to generalized $\{1,2\}$-inverses and specialized forms, revealing the underlying structure of established randomized linear algebra algorithms, including randomized SVD, the Nyström approximation, and CUR decomposition. We demonstrate applications in sparse sensor placement and effective resistance estimation. For the latter, we provide a rigorous quantitative analysis of an approximation scheme, establishing that it always underestimates the true resistance and deriving a worst-case spectral bound on the error of resistance differences.

Figures (6)

• Figure 1: $A\boldsymbol{x}_\text{\footnotesize ${\boldsymbol{r}}$}=\boldsymbol{b}$ is in the column space of $A$ and $A\boldsymbol{x}_\text{\footnotesize ${\boldsymbol{n}}$}=\boldsymbol{0}$. The complete solution to $A\boldsymbol{x}=\boldsymbol{b}$ is $\boldsymbol{x}=$one $\boldsymbol{x}_\text{\footnotesize ${\boldsymbol{r}}$}\,+$any $\boldsymbol{x}_\text{\footnotesize ${\boldsymbol{n}}$}$.
• Figure 2: The four subspaces for $A^{\pmb{+}}$ are the four subspaces for $A^\mathrm{T}$.
• Figure 3: Given a full-rank decomposition $A=CR$, matrix $R$ maps $\hbox{C}(A^T)$ into $\hbox{C}(C^T)$ and matrix $C^T$ maps $\hbox{C}(A)$ into $\hbox{C}(R)$ if and only if $(CR)^+=R^+C^+$.
• Figure 4: The pseudoinverse $A^+=(CR)^+=(C^+CR)^+(CRR^+)^+$ decomposes into the product of the pseudoinverse of $R$ projected on the row space of $C^T$ and the pseudoinverse of $C^T$ projected on the column space of $R$.
• Figure 5: When $A^g$ is a generalized inverse for which only the first Penrose identity holds, or $1$-inverse for $A$, the fundamental subspaces interact in a nontrivial way. The provided example demonstrates the case in which $\mathrm{rank}(A^T) < \mathrm{rank}(A^g)$ and there are vectors in the left nullspace $\hbox{N}(A^T)$ that $A^g$ maps to nontrivial vectors in the column space $\hbox{C}(A^g)$.

Theorems & Definitions (16)

• Theorem 1
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:rol-formula']}
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• proof