Woodbury matrix identity

The Woodbury matrix identity, named after Max A. Woodbury, states that the inverse of a rank -k correction can be a matrix as a rank -k correction of words. Accessible are the names Sherman Morrison - Woodbury formula or just Woodbury formula. However, the equation has been mentioned before Woodbury's report.

The Woodbury equation is

Where A, U, C and V denote matrices in the correct format. Specifically, A is a matrix, U is a matrix, C is a matrix and V is a matrix.

In the special case, the equation also Sherman - Morrison formula is called. Where C is the identity matrix I, the matrix is often called capacitance matrix.

Application

The identity is useful in many numerical calculations, where A- 1 is already calculated and (A UCV ) -1 is required. By the inverse of A, it is only necessary, the inverse of C -1 VA -1U calculated. If C is much smaller dimension than A, which is much more efficient than A UCV to invert directly.

The formula is also used in the derivation of the efficient space representations of the quasi -Newton method.