Generalized Inverses in Linear Algebra Systems
Summary
Generalized inverses extend the notion of the classical matrix inverse to singular or non-square matrices, enabling solutions of linear systems that lack unique or direct inverses. By relaxing one or more of the defining algebraic conditions, these inverses—among them the Moore–Penrose, Drazin, core and Bott–Duffin inverses—provide systematic ways to characterise solvability, compute least-squares solutions and analyse stability properties. Their theory unifies decomposition methods for column and null spaces with functional calculus for singular operators. From constrained regression and model reduction in statistics to control theory, signal processing and network analysis, generalized inverses have become indispensable tools for both theoretical insight and practical computation. Ongoing research balances algebraic characterisations, efficient numerical schemes and broad applicability to time-varying systems, high-dimensional data and structured operators.
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Technical terms
Generalized inverse: A matrix that satisfies a relaxed set of inversion equations when the original is singular or rectangular, allowing system solutions in least-squares or constrained senses.
Moore–Penrose inverse: The unique generalized inverse that fulfils four symmetry and orthogonality conditions, providing minimum-norm least-squares solutions for inconsistent systems.
Drazin inverse: A generalised inverse for square matrices with non-trivial Jordan blocks, used to study singular dynamical systems and index-related spectral properties.
Core inverse: An inner inverse defined by compatibility with the original matrix’s range and null space, particularly suited to constrained optimisation problems.
Bott–Duffin inverse: A projection-based generalised inverse arising in network theory and circuit analysis, characterised by complementary range and null-space projections.
References
- Strong B-T inverse. Heliyon (2024).
- Determinantal Representations of the Core Inverse and Its Generalizations with Applications. Journal of Mathematics (2019).
- The Moore–Penrose inverse: a hundred years on a frontline of physics research. The European Physical Journal H (2021).
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