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. Note that if B is invertible, the two B terms flanking the quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in
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
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
This is useful if one has a matrix with a known inverse A−1 and one needs to invert matrices of the form A+uvT quickly.
If we set A = Ip and B = Iq, we get
In particular, if q = 1, then
- Woodbury matrix identity
- Sherman-Morrison formula
- Invertible matrix
- Matrix determinant lemma
- For certain cases where A is singular and also Moore-Penrose pseudoinverse, see 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
- Moore-Penrose pseudoinverse#Updating the pseudoinverse