SWhat changed
Penrose characterized a unique generalized inverse X for every finite complex rectangular matrix A by AXA=A, XAX=X, (AX)*=AX and (XA)*=XA, with * the conjugate transpose. He gave an existence and uniqueness proof and used generalized inverses to express solvability and all solutions of AXB=C, as well as matrix projectors and a partial-isometry spectral decomposition.
Scientific scope
Penrose characterized a unique generalized inverse X for every finite complex rectangular matrix A by AXA=A, XAX=X, (AX)*=AX and (XA)*=XA, with * the conjugate transpose. He gave an existence and uniqueness proof and used generalized inverses to express solvability and all solutions of AXB=C, as well as matrix projectors and a partial-isometry spectral decomposition. The theorem concerns finite complex matrices, including singular, rectangular and zero matrices; arbitrary rings are explicitly deferred. The AXB=C formula is conditional on its stated compatibility relation and does not imply every right-hand side is exactly solvable. Continuity is asserted at fixed rank, not across rank changes. This record does not assign first priority for generalized inverses or singular-value decomposition, nor include the separate 1956 least-squares paper.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The theorem concerns finite complex matrices, including singular, rectangular and zero matrices; arbitrary rings are explicitly deferred. The AXB=C formula is conditional on its stated compatibility relation and does not imply every right-hand side is exactly solvable. Continuity is asserted at fixed rank, not across rank changes. This record does not assign first priority for generalized inverses or singular-value decomposition, nor include the separate 1956 least-squares paper.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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