Home Statistical Dictionary Moore-Penrose Pseudoinverse

Moore-Penrose Pseudoinverse

The unique generalized inverse satisfying four additional defining conditions, giving the minimum-norm least-squares solution for any matrix -- the tool that lets regression still produce a well-defined, reproducible coefficient estimate even when the design matrix is rank-deficient.

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In Plain English

The Moore-Penrose pseudoinverse is the 'best' generalized inverse in a precise sense: among all the possible generalized inverses a rank-deficient matrix could have, it's the unique one that also gives the smallest-possible-size solution to a least squares problem. That extra uniqueness is exactly why software built on the Moore-Penrose pseudoinverse gives the same, reproducible answer every time for an otherwise ambiguous rank-deficient regression, rather than an answer that depends on which arbitrary generalized inverse happened to be used.

Definition

The Moore-Penrose pseudoinverse $A^{+}$ of a matrix $A$ is the unique matrix satisfying all four Penrose conditions, $A A^{+} A = A$ , $A^{+} A A^{+} = A^{+}$ , and both $(A A^{+})^{⊤} = A A^{+}$ and $(A^{+} A)^{⊤} = A^{+} A$ (both products are symmetric), and, unlike an arbitrary generalized inverse, exists and is unique for every matrix, computed in practice from the singular value decomposition, $A^{+} = V Σ^{+} U^{⊤}$ . For a full-column-rank matrix $X$ , the pseudoinverse coincides with the ordinary formula $X^{+} = (X^{⊤} X)^{- 1} X^{⊤}$ , and for a rank-deficient design matrix, $\hat{β} = X^{+} y$ gives the specific least squares solution with the smallest Euclidean norm among all coefficient vectors that fit the data equally well, resolving the non-uniqueness that an ordinary generalized inverse leaves open.

Formula

A^{+} = V Σ^{+} U^{⊤}

Notation

A^{+}

—the Moore-Penrose pseudoinverse of A

U, Σ, V

—the singular value decomposition factors of A

Properties

The Moore-Penrose pseudoinverse is the unique generalized inverse satisfying all four Penrose conditions -- this extra structure is exactly what removes the non-uniqueness that plagues an arbitrary generalized inverse, making the pseudoinverse the standard default choice whenever software needs to produce one specific, reproducible answer for a rank-deficient linear system.
For a rank-deficient least squares problem, $\hat{β} = X^{+} y$ gives the minimum-norm solution, among the infinitely many coefficient vectors that fit the observed data equally well, it selects the one with the smallest sum of squared coefficients -- a specific, well-defined tie-breaking rule, not an arbitrary one.
When a matrix is invertible, its pseudoinverse reduces exactly to its ordinary inverse, and when a design matrix has full column rank, its pseudoinverse reduces exactly to the familiar $(X^{⊤} X)^{- 1} X^{⊤}$ formula -- the pseudoinverse is a strict generalization that agrees with the tools already in use whenever those tools are well-defined.

At a Glance

Also calledthe pseudoinverse

Uniquenessunique -- unlike a general generalized inverse

Computed viathe singular value decomposition

Givesthe minimum-norm least squares solution for a rank-deficient system

Last updated September 8, 2026← Back to the Dictionary