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. 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.[1]

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

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

A more general formula exists when B is singular and possibly even non-square:[1]

Formulas also exist for certain cases in which A is singular.[2]

Verification[edit]

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.[3]

Special cases[edit]

First[edit]

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


Second[edit]

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.

Third[edit]

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.

See also[edit]

References[edit]

  1. ^ a b Henderson, H. V., and Searle, S. R. (1981), "On deriving the inverse of a sum of matrices", SIAM Review 23, pp. 53-60 doi:10.1137/1023004 [1].
  2. ^ 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 MR1152773
  3. ^ Gilbert Strang (2003). Introduction to Linear Algebra (3rd ed.). Wellesley-Cambridge Press: Wellesley, MA. ISBN 0-9614088-9-8.