Woodbury matrix identity
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In mathematics (specifically linear algebra), the Woodbury matrix identity 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. Alternative names for this formula are the matrix inversion lemma and the Sherman-Morrison-Woodbury formula.
Explicitly, the Woodbury matrix identity is
- <math> (A+UCV)^{-1} = A^{-1} - A^{-1}U (C^{-1}+VA^{-1}U)^{-1} VA^{-1}, <math>
where A, U, C and V all denote matrices of the correct size. Specifically, A is n-by-n, U is n-by-k, C is k-by-k and V is k-by-n.
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Applications
This identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute (A + UCV)−1. With the inverse of A available, it is only necessary to find the inverse of C−1+VA−1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A+UCV directly.
This is applied, e.g., in the Kalman filter and other least-squares estimation methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter.
See also
References
- Gene H. Golub and Charles F. Van Loan, Matrix computations (3rd ed.), page 50, John Hopkins University Press, 1996.
External links
- Some matrix identities (http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/identity.html)it:Identità di Woodbury