In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury 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, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.
The Woodbury matrix identity is
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. This can be derived using blockwise matrix inversion.
In the special case where C is the 1-by-1 unit matrix, this identity reduces to the Sherman–Morrison formula. In the special case when C is the identity matrix I, the matrix is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.
Read more about Woodbury Matrix Identity: Derivation Via Blockwise Elimination, Derivation From LDU Decomposition, Direct Proof, Applications
Famous quotes containing the words matrix and/or identity:
“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
—Margaret Atwood (b. 1939)
“One of the most highly valued functions of used parents these days is to be the villains of their children’s lives, the people the child blames for any shortcomings or disappointments. But if your identity comes from your parents’ failings, then you remain forever a member of the child generation, stuck and unable to move on to an adulthood in which you identify yourself in terms of what you do, not what has been done to you.”
—Frank Pittman (20th century)