In mathematics, 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 sizes. 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. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category.
Discussion
To prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrixI, we obtain another identity which is a bit simpler: To recover the original equation from this reduced identity, set and. This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from thus, and similarly The second identity is the so-called push-through identity that we obtain from after multiplying by on the right and by on the left.
Special cases
When are vectors, the identity reduces to the Sherman–Morrison formula. In the scalar case it is simply
Inverse of a sum
If p = q and U = V = Ip is the identity matrix, then Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity Another useful form of the same identity is which has a recursive structure that yields This form can be used in perturbative expansions where B is a perturbation of A.
Variations
Binomial inverse theorem
If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals and the rank of the latter cannot exceed the rank of B. Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with which results in the original Woodbury identity. A variation for when B is singular and possibly even non-square: Formulas also exist for certain cases in which A is singular.
Derivations
Direct proof
The formula can be proven by checking that times its alleged inverse on the right side of the Woodbury identity gives the identity matrix:
Alternative proofs
First consider these useful identities, Now, Deriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problem Expanding, we can see that the above reduces to which is equivalent to. Eliminating the first equation, we find that, which can be substituted into the second to find. Expanding and rearranging, we have, or. Finally, we substitute into our, and we have. Thus, We have derived the Woodbury matrix identity. We start by the matrix By eliminating the entry under the A we get Likewise, eliminating the entry above C gives Now combining the above two, we get Moving to the right side gives which is the LDU decomposition of the block matrix into an upper triangular, diagonal, and lower triangular matrices. Now inverting both sides gives We could equally well have done it the other way i.e. Now again inverting both sides, Now comparing elements of the RHS of and above gives the Woodbury formula
Applications
This identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute −1. With the inverse of A available, it is only necessary to find the inverse of C−1 + VA−1U 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 + UCV directly. A common case is finding the inverse of a low-rank update A + UCV of A, or finding an approximation of the inverse of the matrix A + 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 is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.
