Weighted Pseudoinverses: Theory & Applications

• Weighted pseudoinverses are generalized inverses that incorporate nontrivial quadratic forms via weight matrices or operators to address non-Euclidean optimization problems.
• They extend the traditional Moore–Penrose inverse to complex operator settings, thereby unifying methods in statistics, control theory, and signal processing.
• Applications include solving weighted least squares, generalized Procrustes problems, and structured matrix computations with efficient algorithms for large-scale data.

Weighted pseudoinverses generalize the Moore–Penrose pseudoinverse by incorporating nontrivial quadratic forms or bilinear structures—typically via “weight” operators or matrices—into the algebra of generalized inverses. This theoretical framework subsumes a wide spectrum of generalized inverses, notably the weighted Moore–Penrose inverse, weighted Drazin-type inverses, BT-, core-EP-, and group inverses, and their regularized variants relevant for structured least squares and generalized Procrustes problems. Weighted pseudoinverses are fundamental in the analysis of operator equations on Hilbert spaces and Hilbert C*-modules, as well as finite-dimensional matrix problems in statistics, control, signal processing, and numerical linear algebra.

1. Foundational Concepts and Core Definitions

Weighted pseudoinverses are constructed to minimize weighted norms or solve generalized least squares problems subject to linear constraints in a non-Euclidean geometry. Let $A: H \to K$ be a bounded linear operator between Hilbert spaces (or, more generally, an adjointable operator on Hilbert C*-modules), and let $M \in L(K)$, $N \in L(H)$ be self-adjoint, invertible “weight” operators. The $M$-$N$ weighted Moore–Penrose inverse of $A$, denoted $A^\dag_{M,N}$, is the unique operator $X$ satisfying the four weighted Penrose equations:

• $A X A = A$,
• $X A X = X$,
• $M \in L(K)$0,
• $M \in L(K)$1.

Existence of $M \in L(K)$2 requires $M \in L(K)$3 to have closed range; uniqueness follows from these equations (Xu et al., 2010, Xu, 14 Feb 2025). In the matrix setting, the weights $M \in L(K)$4 and $M \in L(K)$5 are positive-definite Hermitian matrices, but recent generalizations allow them to be arbitrary invertible, self-adjoint operators—even indefinite—on module frameworks (Xu, 14 Feb 2025).

Specialized weighted inverses (core-EP, Drazin, BT, group, etc.) are defined by systems of algebraic equations involving suitable powers of $M \in L(K)$6 and associated projectors, and further characterized in terms of prescribed range or nullspace conditions. The “$M \in L(K)$7-weighted” case refers to a single weight matrix $M \in L(K)$8 (often nonsingular, not necessarily Hermitian), entering quadratic or higher-order defining equations. For example, the $M \in L(K)$9-weighted Moore–Penrose inverse for $N \in L(H)$0, $N \in L(H)$1 solves: $N \in L(H)$2 (Chowdhry et al., 2023).

2. Weighted Least Squares, Procrustes Problems, and Operator Theory

Weighted pseudoinverses naturally arise in weighted least squares and Procrustes problems. Given $N \in L(H)$3 bounded operators on a separable Hilbert space $N \in L(H)$4, and a positive operator $N \in L(H)$5 with $N \in L(H)$6 in the Schatten class $N \in L(H)$7 for some $N \in L(H)$8, the problem

$N \in L(H)$9

admits a solution exactly when the compatibility condition $M$0 holds. The minimizers (“$M$1-inverses”) are precisely the solutions of the weighted normal equation $M$2. In the $M$3 case (Hilbert–Schmidt norm), this reduces to a standard Euclidean projection; in other Schatten norms, it is a nontrivial nonlinear projection (Contino et al., 2016).

The explicit weighted pseudoinverse (when $M$4 is invertible) is

$M$5

and the set of all minimizers is affine—uniqueness is guaranteed when $M$6 is injective on $M$7 (Contino et al., 2016).

This paradigm extends to operator algebras, including Hilbert C*-modules, via generalizations of the Penrose equations and “weighted adjoint” maps, where the analysis requires additional care due to the possible indefiniteness of the underlying quadratic forms or the lack of orthogonal complements (Xu et al., 2010, Xu, 14 Feb 2025).

3. Algebraic Structure and Explicit Formulas

Weighted pseudoinverses admit closed-form formulas in terms of $M$8, $M$9, $N$0, and $N$1. A fundamental result (Xu, 14 Feb 2025) gives

$N$2

with $N$3, $N$4. When $N$5 and $N$6 are indefinite, there always exist positive-definite “effective” weights $N$7, $N$8 so that $N$9, showing that indefinite-weighted pseudoinverses reduce to ordinary weighted theory for positive weights (Xu, 14 Feb 2025).

Block formulas generalize to partitioned and operator-valued cases. For block operators on Hilbert C*-modules, the weighted Moore–Penrose inverse of a $A$0 or $A$1 matrix can be represented explicitly in terms of the unweighted inverse of certain induced blocks and the weights (Xu et al., 2010).

Other weighted pseudoinverses (weighted core-EP, BT, Drazin, etc.) are built from spectral decompositions and nilpotent/cyclic structures; canonical forms result from block/unitary/SVD-based representations—essential, for example, for the $A$2-weighted $A$3-BT inverse: $A$4 where $A$5 projects onto $A$6, and $A$7 interpolates between distinct inverse types. For $A$8 the $A$9-weighted Moore–Penrose is obtained, for $A^\dag_{M,N}$0 the BT inverse, and for $A^\dag_{M,N}$1 the weighted core-EP inverse (Ferreyra et al., 2024).

Weighted Drazin, weak-group, and related inverses are characterized by higher-order polynomial identities and outer-inverse properties, often admitting minimal-rank realizations and explicit block formulas under core-nilpotent decompositions (Senapati et al., 30 Dec 2025, Gao et al., 2024, Chowdhry et al., 2023).

4. Uniqueness, Existence, and Structural Properties

Existence of weighted pseudoinverses typically requires closed range (operators) or full-rank (matrices) conditions, plus compatibility of the rank of $A^\dag_{M,N}$2 with $A^\dag_{M,N}$3 (or associated polynomials in $A^\dag_{M,N}$4). For the $A^\dag_{M,N}$5-weighted $A^\dag_{M,N}$6-inverse (generalizing the core inverse), $A^\dag_{M,N}$7 if and only if $A^\dag_{M,N}$8, and uniqueness holds precisely when the relevant index is $A^\dag_{M,N}$9 or $X$0 (Chowdhry et al., 2023).

Weighted DMP/MPD inverses admit minimal-rank characterizations and order-laws: for instance, under commutativity, the $X$1-weighted Drazin inverse of $X$2 factors as $X$3 (Senapati et al., 30 Dec 2025). Projection and rank-based criteria are fundamental throughout: the range and nullspace of weighted pseudoinverses can often be prescribed explicitly (Ferreyra et al., 2024, Senapati et al., 30 Dec 2025, Gao et al., 2024).

Continuity and perturbation theory show that the weighted pseudoinverse depends continuously on $X$4, with norm estimates available for finite-dimensional and infinite-dimensional cases, and explicit perturbation bounds are derived for minimal-rank Drazin-type inverses (Xu, 14 Feb 2025, Senapati et al., 30 Dec 2025).

5. Computational Methods and Applications

Efficient algorithms for computing weighted pseudoinverses have been developed, with particular emphasis on structured/sparse or rational/polynomial matrix data. Partitioning methods, such as variants of the Greville–Wang recursive algorithm, enable symbolic computation of the weighted Moore–Penrose inverse for rational or polynomial matrices (Tasić et al., 2011). The complexity is $X$5 for matrices of size $X$6 and polynomial degree $X$7.

For large-scale problems, iterative algorithms based on generalized Golub–Kahan bidiagonalization and generalized LSQR efficiently compute minimal-norm solutions to weighted and constrained least squares problems. These methods leverage Hilbert space interpretations and operator pseudoinverses to exploit the structure of the constraint/penalty weights, and, when combined with GSVD, yield robust and scalable solvers for high-dimensional data (Li, 2024).

Weighted pseudoinverses play a critical role in regularized regression, statistical inverse problems, model reduction, and control—the generalized least squares (GLS) solution is naturally given by the action of a weighted pseudoinverse, and statistical properties depend directly on the structure of the weight (Li, 2024, Contino et al., 2016).

6. Theoretical Unification and Generalizations

Recent advances unify the theory of weighted pseudoinverses across a broad spectrum of generalized inverses—Drazin, group, BT, core-EP, and others—by parametrizing families such as the $X$8-weighted $X$9-BT inverse or the $A X A = A$0-weighted m-weak group MP inverse. These families admit canonical representations via block-unitary or core-nilpotent decompositions, provide closed-form solutions to structured matrix and operator equations, and clarify interrelations among existing inverse notions (Ferreyra et al., 2024, Gao et al., 2024, Chowdhry et al., 2023). The outer-inverse formalism, projection formulas, and continuity properties extend naturally to operator-theoretic and module settings (Xu, 14 Feb 2025, Xu et al., 2010).

This general theory reveals that weighted pseudoinverses are not merely technical extensions, but are organizing principles unifying constrained optimization in geometry, statistics, and operator algebras. By controlling the algebraic and analytic properties of inverses through adjustable weights, one can obtain family-wide structural, continuity, and computational guarantees across matrix and operator frameworks.