Sherman–Morrison formula

The Sherman–Morrison formula expresses the inverse of a rank-one update to an invertible matrix using the original inverse and two vectors. It applies when its scalar denominator is nonzero.

Rank-one matrix: A matrix whose column space has dimension one. The update is an outer product of two vectors, so it has rank at most one.
Invertible matrix: A square matrix that has a two-sided multiplicative inverse. The identity starts with an invertible matrix and uses its inverse throughout.
Recursive least squares: An adaptive estimation method that updates least-squares parameters as observations arrive. Its covariance inverse can be updated efficiently when a new observation adds one rank-one term.
Banachiewicz formula: A block-matrix inverse formula expressed through Schur complements. It handles partitioned block changes rather than a single outer-product update.
Matrix inverse: A matrix that multiplies a given square matrix to produce the identity matrix. The formula computes the inverse after perturbing an already invertible matrix.
Identity matrix: A square matrix with ones on its diagonal and zeros elsewhere. Multiplying the proposed updated inverse by the updated matrix must yield this matrix.
Kalman filter: A recursive algorithm that estimates a changing state from noisy measurements. Rank-one inverse updates can streamline covariance calculations in measurement-update steps.
Matrix inversion: The general operation of finding a matrix’s multiplicative inverse. Recomputing an inverse directly is the broad alternative the update identity can avoid.
Matrix determinant lemma: An identity relating the determinant of a low-rank update to the determinant of the original matrix. Its scalar factor matches the denominator that determines whether the updated matrix is invertible.
Matrix multiplication: An operation combining compatible matrices by taking row-by-column products. Direct verification expands the product of the updated matrix and its proposed inverse.

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