Updatematrix

Updatematrix is a term used in linear algebra and numerical analysis to describe a matrix that encodes incremental changes to another matrix or to a system state. The idea is to separate the original matrix A from the perturbation U that represents updates, so that the updated matrix can be written as A_new = A_old + U or as A + δA, where δA is the update.

Representations of an updatematrix can vary. In many cases U is sparse, diagonal, or low-rank, such as U = P Q^T for a rank-k update. This enables efficient application of updates without reconstructing the full matrix. Updates may be applied explicitly or applied in a matrix-free fashion, often in conjunction with existing factorizations or iterative solvers.

Applications of the updatematrix concept appear across several areas. In numerical linear algebra, rank-one or low-rank updates are used to modify Cholesky, LU, or QR factorizations without rebuilding them from scratch. In online learning and streaming data, updatematrices allow model or covariance matrices to be adjusted as new data arrives. In optimization and Kalman filtering, updates to system matrices or covariances are common, and the updatematrix provides a compact way to record these changes.

Computational considerations include the cost of applying the update, numerical stability, and the need for reconditioning when updates accumulate. If updates remain low-rank or sparse, the benefits include reduced computation and memory usage compared to rebuilding from scratch.

See also perturbation theory, rank-one update, Sherman–Morrison–Woodbury formula, and matrix factorization.