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Sherman–Morrison Formula

The Sherman–Morrison formula gives the exact inverse of an invertible matrix after a rank-one outer-product update, provided one scalar denominator is nonzero.

Version
v3 · 2026-09-06 · History
Domain-specific #
2770
Origin domain
linear algebra
Subdomain
matrix analysis
Aliases
Sherman–Morrison identity, Rank-one inverse update formula

Core Idea

The Sherman–Morrison formula is the exact rank-one inverse-update identity. Let (A) be an invertible \(n\times n\) matrix, and let (u) and (v) be (n)-component column vectors. If

\[ \alpha = 1+v^{\mathsf T}A^{-1}u \neq 0, \]

then \(A+uv^{\mathsf T}\) is invertible and

\[ (A+uv^{\mathsf T})^{-1} =A^{-1}-\frac{A^{-1}uv^{\mathsf T}A^{-1}} {1+v^{\mathsf T}A^{-1}u}. \]

Over complex matrices the same relation is commonly written with a conjugate transpose, (uv^*), when that is the intended update. More abstractly, \(v^{\mathsf T}\) is a row covector. No symmetry or positive definiteness is required. The assumptions are an invertible baseline, a rank-at-most-one outer-product modification, and a nonzero scalar denominator.

Scope of Application

Sherman–Morrison belongs to linear algebra and numerical computation wherever a sequence of closely related linear systems differs by one rank-one term. The historical Sherman–Morrison papers treated changes to matrix elements, rows, or columns; Bartlett stated a general inverse adjustment in discriminant analysis. Hager's SIAM review documents the wider matrix-modification family across statistics, networks, structural analysis, optimization, partial differential equations, and related applications.

Clarity

The scalar \(\alpha=1+v^{\mathsf T}A^{-1}u\) clarifies three questions at once. First, it decides whether the updated matrix is invertible. Second, it scales the entire inverse correction. Third, its magnitude warns when the update approaches singularity. If \(\alpha=0\), then

\[ (A+uv^{\mathsf T})A^{-1}u =u+u(v^{\mathsf T}A^{-1}u)=u\alpha=0. \]

Manages Complexity

The formula compresses a global change into low-dimensional information. Although changing (A) changes every entry of its inverse in general, a rank-one perturbation changes that inverse by another rank-one matrix. The calculation can be organized around

\[ z=A^{-1}u,\qquad w^{\mathsf T}=v^{\mathsf T}A^{-1},\qquad \alpha=1+v^{\mathsf T}z, \]

Abstract Reasoning

The identity follows from factoring the update:

\[ A+uv^{\mathsf T}=A(I+A^{-1}uv^{\mathsf T}). \]

Let (z=A^{-1}u). Since \((zv^{\mathsf T})^2=z(v^{\mathsf T}z)v^{\mathsf T}\), every positive power of the rank-one product remains a scalar multiple of the same outer product. One can therefore seek an inverse of \(I+zv^{\mathsf T}\) in the form \(I-\beta zv^{\mathsf T}\). Multiplication gives the identity when

Knowledge Transfer

The formula transfers literally across statistics, optimization, control, network computation, and scientific modeling because the same matrix relation survives: an invertible baseline plus one outer product. The domain objects differ—an observation, basis-column replacement, covariance contribution, constraint, edge term, or physical coupling—but the algebraic roles remain (A,u,v,alpha), and the same correction follows.

What transfers outside matrix mathematics is only analogy: reuse a solved baseline when a modification is low-dimensional. That more general heuristic belongs to Perturbation, Refinement, or related catalog abstractions.

Relationships to Other Abstractions

Local relationship map for Sherman–Morrison FormulaParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Sherman–MorrisonFormulaDOMAINDomain-specific abstraction: Matrix — presupposesMatrixDOMAIN

Current abstraction Sherman–Morrison Formula Domain-specific

Parents (1) — more general patterns this builds on

  • Sherman–Morrison Formula presupposes Matrix Domain-specific

    Perturbation. (uv^{\mathsf T}) modifies a baseline matrix.

Hierarchy paths (5) — routes to 5 parentless roots

Neighborhood in Abstraction Space

Sherman–Morrison Formula sits in a sparse region of the domain-specific corpus (93rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08