Sample matrix inversion

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title: "Sample matrix inversion" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["covariance-and-correlation", "filter-theory"] topic_path: "general/covariance-and-correlation" source: "https://en.wikipedia.org/wiki/Sample_matrix_inversion" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

Sample matrix inversion (or direct matrix inversion) is an algorithm that estimates weights of an array (adaptive filter) by replacing the correlation matrix R with its estimate. Using K N-dimensional samples X_1, X_2,\dots,X_K, an unbiased estimate of R_{X}, the N \times N correlation matrix of the array signals, may be obtained by means of a simple averaging scheme: :\hat{R}{X} = \frac{1}{K} \sum\limits{k=1}^K X_k X^H_k, where H is the conjugate transpose. The expression of the theoretically optimal weights requires the inverse of R_{X}, and the inverse of the estimates matrix is then used for finding estimated optimal weights.

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