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 $p\times p,\ p\times q,\ q\times q,\ q\times p$ respectively, then

$\left(A+UBV\right)^{-1}= A^{-1} - A^{-1}UB\left(B+BVA^{-1}UB\right)^{-1}BVA^{-1}$

provided $A\!$ and $B + BVA^{-1}UB\!$ 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

$\left(A+UBV\right)^{-1}= A^{-1} - A^{-1}U\left(B^{-1}+VA^{-1}U\right)^{-1}VA^{-1}$

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

[edit] Verification

First notice that

$\left(A + UBV\right) A^{-1}UB = UB + UBVA^{-1}UB = U \left(B + BVA^{-1}UB\right).$

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

$\left(A + UBV\right) \left( A^{-1} - A^{-1}UB\left(B + BVA^{-1}UB\right)^{-1}BVA^{-1} \right)$

$= I_p + UBVA^{-1} - U \left(B + BVA^{-1}UB\right) \left(B + BVA^{-1}UB\right)^{-1}BVA^{-1}$

$= I_p + UBVA^{-1} - U BVA^{-1} = I_p \!$

which verifies that it is the inverse.

So we get that -- if $A^{-1}\!$ and $\left(B + BVA^{-1}UB\right)^{-1}$ exist, then $\left(A + UBV\right)^{-1}$ exists and is given by the theorem above.

[edit] Special cases

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

$\left(A+B\right)^{-1} = A^{-1} - A^{-1}B\left(B+BA^{-1}B\right)^{-1}BA^{-1}.$

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