Binomial inverse theorem
||It has been suggested that this article be merged into Woodbury matrix identity. (Discuss) Proposed since January 2016.|
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 B(I+VA—1UB) 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 (B−1)−1, which results in
A more general formula exists when B is singular and possibly even non-square:
Formulas also exist for certain cases in which A is singular.
First notice that
Now multiply the matrix we wish to invert by its alleged inverse:
which verifies that it is the inverse.
So we get that if A−1 and exist, then exists and is given by the theorem above.
If p = q and U = V = Ip is the identity matrix, then
Remembering the identity
we can also express the previous equation in the simpler form as
Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity
If B = Iq is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written vT. Then the theorem implies the Sherman-Morrison formula:
This is useful if one has a matrix A with a known inverse A−1 and one needs to invert matrices of the form A+uvT quickly for various u and v.
If we set A = Ip and B = Iq, we get
In particular, if q = 1, then
which is a particular case of the Sherman-Morrison formula given above.
- Henderson, H. V., and Searle, S. R. (1981), "On deriving the inverse of a sum of matrices", SIAM Review 23, pp. 53-60.
- Kurt S. Riedel, "A Sherman—Morrison—Woodbury Identity for Rank Augmenting Matrices with Application to Centering", SIAM Journal on Matrix Analysis and Applications, 13 (1992)659-662, doi:10.1137/0613040 preprint MR 1152773
- Gilbert Strang (2003). Introduction to Linear Algebra (3rd ed.). Wellesley-Cambridge Press: Wellesley, MA. ISBN 0-9614088-9-8.