Woodbury matrix identity

The Woodbury matrix identity expresses the inverse of a low-rank update to an invertible matrix using the original inverse and a smaller matrix inverse.

Matrix inverse: A matrix that multiplies another matrix to produce the identity matrix. The identity updates this inverse rather than recomputing it from scratch.
Rank (linear algebra): The dimension of the vector space spanned by a matrix’s columns or rows. The update rank determines the size of the reduced matrix to invert.
Kalman filter: An algorithm that estimates a changing state from noisy measurements using recursive linear updates. Measurement updates often reduce to efficient covariance-inverse updates.
Direct matrix inversion: Computing a matrix inverse from the matrix itself rather than updating a known inverse. Woodbury can avoid this costly computation when the change has low rank.
Sherman–Morrison formula: A formula for updating a matrix inverse after a rank-one change. It is the rank-one special case of the Woodbury identity.
Invertible matrix: A square matrix that has a two-sided multiplicative inverse. The identity assumes the original matrix is invertible.
Gaussian process: A probability distribution over functions specified by jointly Gaussian values at finite sets of inputs. Low-rank covariance approximations use Woodbury to accelerate inference.
Sherman–Morrison–Woodbury formula: A general name for inverse-update identities including rank-one and low-rank cases. This broader family places Woodbury alongside its rank-one specialization.
Matrix determinant lemma: An identity relating the determinant of a low-rank matrix update to a smaller determinant. It accompanies Woodbury’s inverse update and uses a similar reduced matrix.
Identity matrix: A square matrix with ones on the diagonal and zeros elsewhere. It appears in the reduced matrix whose inverse corrects the original inverse.

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