Woodbury's identity
In mathematics (specifically linear algebra), the Woodbury matrix identity is the following formula for n×n matrices:
Applications
Often, the matrix A is numerically large, while UCV is small, a perturbation of A. If furthermore the dimensionality of C is much smaller than that of A, we can get away with inverting only two matrices of this smaller dimension, while re-using the pre-existing inverse A−1.
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.
Proof
Consider the following partitioned matrix equation:
Write this out into four equations:
Of these, only the first and third are needed.
Add the third to the first after multiplying by  :
Then, subtract the first from the third after multiplying by  :
Back substitute into the first equation:
Now we have two different expressions for the sub-matrix  that should be identical. Thus we obtain:
Which completes the proof.
See also:
External links
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