DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations

Definition

Let $F$ be $R$ or $C$ , and let $A \in M_{m \times n} (F)$ . A matrix $B \in M_{n \times m} (F)$ is a Moore--Penrose pseudoinverse of $A$ when

$A B A = A, B A B = B,$

and the two square products are self-adjoint:

(A B)^{*} = A B, (B A)^{*} = B A .

When such a matrix exists and is unique, it is denoted by $A^{+}$ .

The four displayed relations are the Penrose equations. They are written in the matrix product and adjoint conventions of Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes and In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix.

Depends on

Used by

Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse Theorem

Dependency tree · two levels

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Sources

Roger Penrose, A generalized inverse for matrices (standard reference, not scraped)

The Moore--Penrose pseudoinverse A+A+ as the solution of the four Penrose equations