Binomial inverse theorem
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In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways.
If 
are matrices of sizes 
respectively, then
provided 
and 
are nonsingular. Note that if 
is invertible, the two terms flanking the quantity inverse in the right-hand side can be replaced with 
, which results in
This is the matrix inversion lemma, which can also be derived using matrix blockwise inversion.
[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 
and 
exist, then 
exists and is given by the theorem above.
[edit] Special cases
If 
and 
is the identity matrix, then
If 
is the identity matrix and 
, then U is a column vector, written u, and V is a row vector, written vT. 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 + uvT quickly.
If we set 
and , we get
In particular, if , then
