Editing Woodbury matrix identity (section)

== Applications ==

This identity is useful in certain numerical computations where ''A''<sup>&minus;1</sup> has already been computed and it is desired to compute (''A''&nbsp;+&nbsp;''UCV'')<sup>&minus;1</sup>.  With the inverse of ''A'' available, it is only necessary to find the inverse of ''C''<sup>−1</sup>&nbsp;+&nbsp;''VA''<sup>−1</sup>''U'' 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''&nbsp;+&nbsp;''UCV'' directly. A common case is finding the inverse of a low-rank update ''A''&nbsp;+&nbsp;''UCV'' of ''A'' (where ''U'' only has a few columns and ''V'' only a few rows), or finding an approximation of the inverse of the matrix ''A''&nbsp;+&nbsp;''B'' where the matrix ''B'' can be approximated by a low-rank matrix ''UCV'', for example using the [[singular value decomposition]].

This is applied, e.g., in the [[Kalman filter]] and [[recursive least squares]] 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.

In the case when ''C'' is the identity matrix ''I'', the matrix <math>I+VA^{-1}U</math> is known in [[numerical linear algebra]] and [[numerical partial differential equations]] as the '''capacitance matrix'''.<ref name="hager"/>