Supriya Ghosh (Editor)

Binomial inverse theorem

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

Contents

If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then

( A + U B V ) − 1 = A − 1 − A − 1 U B ( B + B V A − 1 U B ) − 1 B V A − 1

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 + U B V ) − 1 = A − 1 − A − 1 U ( B − 1 + V A − 1 U ) − 1 V A − 1 .

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:

( A + U B V ) − 1 = A − 1 − A − 1 U ( I + B V A − 1 U ) − 1 B V A − 1 .

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

Verification

First notice that

( A + U B V ) A − 1 U B = U B + U B V A − 1 U B = U ( B + B V A − 1 U B ) .

Now multiply the matrix we wish to invert by its alleged inverse:

( A + U B V ) ( A − 1 − A − 1 U B ( B + B V A − 1 U B ) − 1 B V A − 1 ) = I p + U B V A − 1 − U ( B + B V A − 1 U B ) ( B + B V A − 1 U B ) − 1 B V A − 1 = I p + U B V A − 1 − U B V A − 1 = I p

which verifies that it is the inverse.

So we get that if A−1 and ( B + B V A − 1 U B ) − 1 exist, then ( A + U B V ) − 1 exists and is given by the theorem above.

First

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

( A + B ) − 1 = A − 1 − A − 1 B ( B + B A − 1 B ) − 1 B A − 1 .

Remembering the identity

( C D ) − 1 = D − 1 C − 1 ,

we can also express the previous equation in the simpler form as

( A + B ) − 1 = A − 1 − A − 1 ( I + B A − 1 ) − 1 B A − 1 .

Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity

( A + B ) − 1 = A − 1 − ( A + A B − 1 A ) − 1 .

Second

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:

( A + u v T ) − 1 = A − 1 − A − 1 u v T A − 1 1 + v T A − 1 u .

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

If we set A = Ip and B = Iq, we get

( I p + U V ) − 1 = I p − U ( I q + V U ) − 1 V .

In particular, if q = 1, then

( I + u v T ) − 1 = I − u v T 1 + v T u ,

which is a particular case of the Sherman-Morrison formula given above.

References

Binomial inverse theorem Wikipedia
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