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In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury [1] [2] – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.
The matrix determinant lemma performs a rank-1 update to a determinant. Woodbury matrix identity; Quasi-Newton method; Binomial inverse theorem; Bunch–Nielsen–Sorensen formula; Maxwell stress tensor contains an application of the Sherman–Morrison formula.
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:
The Woodbury matrix identity used in linear algebra is named after him. [7] [16] The related Sherman–Morrison formula is a special case of the formula, [17] [18] [19] with the term Sherman-Morrison-Woodbury sometimes used. An early overview of some of its uses has been given by Hager, [20] see also the book "Woodbury Matrix Identity". [21]
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Euler's identity; Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities; Heine's identity; Hermite's identity; Lagrange's identity; Lagrange's trigonometric identities; List of logarithmic identities; MacWilliams identity; Matrix determinant lemma; Newton's identity; Parseval's identity; Pfister's sixteen ...
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For the cases where has full row or column rank, and the inverse of the correlation matrix ( for with full row rank or for full column rank) is already known, the pseudoinverse for matrices related to can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the ...