Woodbury Matrix Identity

The Woodbury Matrix Identity , named after Max A. Woodbury, states that the inverse of a rank correction of a matrix can be expressed as a rank correction of the inverse . The terms Sherman-Morrison-Woodbury formula or just Woodbury formula are also common . But the equation was mentioned before Woodbury's report. ${\displaystyle k}$ ${\displaystyle A}$ ${\displaystyle k}$ ${\displaystyle A^{-1}}$

The Woodbury equation is

${\displaystyle \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}}$,

wherein , , and denote matrices of the correct size. More precisely, a matrix, a matrix, a matrix and a matrix. ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle C}$ ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle n\times n}$ ${\displaystyle U}$ ${\displaystyle n\times k}$ ${\displaystyle C}$ ${\displaystyle k\times k}$ ${\displaystyle V}$ ${\displaystyle k\times n}$

In the special case and , the equation is also called the Sherman-Morrison formula . If the identity matrix , the matrix is often capacitance matrix called. ${\displaystyle k=1}$ ${\displaystyle C=1}$ ${\displaystyle C}$ ${\displaystyle I}$ ${\displaystyle I+VA^{-1}U}$

application

The identity is useful in many numerical calculations that have already been calculated and needed. With the inverse of , it is only necessary to calculate the inverse of . If it has a much smaller dimension than , that is much more efficient than inverting directly. ${\displaystyle A^{-1}}$ ${\displaystyle (A+UCV)^{-1}}$ ${\displaystyle A}$ ${\displaystyle C^{-1}+VA^{-1}U}$ ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle A+UCV}$

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

See also

• Regular matrix

Web links

• Some matrix identities
• Eric W. Weisstein : Woodbury formula . In: MathWorld (English).

Individual evidence

1. ↑ Max A. Woodbury: Inverting modified matrices. In: Memorandum Rept. 42, Statistical Research Group, Princeton University, Princeton NJ 1950, 4pp MR38136
2. ^ Max A. Woodbury: The Stability of Out-Input Matrices. Chicago IL 1949, 5 pages, MR32564
3. ^ ^{A b} William W. Hager: Updating the inverse of a matrix . In: SIAM Review . 31, No. 2, 1989, pp. 221-239. MR 997457 - JSTOR 2030425 . doi : 10.1137 / 1031049 .
4. ^ Nicholas Higham : Accuracy and Stability of Numerical Algorithms , 2nd. Edition, SIAM , 2002, ISBN 978-0-89871-521-7 , p. 258, MR 1927606 .
5. ↑ Byrd Nocedal Schnabel: Representations of quasi-Newton matrices and their use in limited memory methods . In: Mathematical Programming . 63, No. 1, 1994, pp. 129-156.