Binomial inverse theorem

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.

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:

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 ${\displaystyle \left(\mathbf {B} +\mathbf {BVA} ^{-1}\mathbf {UB} \right)^{-1}}$ exist, then ${\displaystyle \left(\mathbf {A} +\mathbf {UBV} \right)^{-1}}$ exists and is given by the theorem above.

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