Binomial inverse theorem

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In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways.

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

This is the matrix inversion lemma, which can also be derived using matrix blockwise inversion.

Contents 1 Verification 2 Special cases 3 See also 4 References

[edit] Verification

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.^[1]

[edit] Special cases

If p = q and U = V = I_p is the identity matrix, then

If B = I_q is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written v^T. Then the theorem implies

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+uv^T quickly.

If we set A = I_p and B = I_q, we get

In particular, if q = 1, then

[edit] See also

• 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

[edit] References

1. ^ Gilbert Strang (2003). Introduction to Linear Algebra (3rd edition ed.). Wellesley-Cambridge Press: Wellesley, MA. ISBN 0-9614088-98.