Thread: identity proof

hey guys,

i need help in proving this identity. i've been working on it for 3 days now

some notation:
A is a matrix;
A^T is A transpose;
A^-1 is A inverse;
v is a vector;
v^-1 is v inverse;
v^T is v transpose;

Assume A is a non singular matrix and v is a vector with comparable dimension. Prove:

(A + vv^T)^-1 = A^-1 - [ (A^-1v)(v^TA^-1) / (1 + v^TA^-1v) ]


Need help please. or at least some tips. Thanks!

Originally Posted by appsmail12

hey guys,

i need help in proving this identity. i've been working on it for 3 days now

some notation:
A is a matrix;
A^T is A transpose;
A^-1 is A inverse;
v is a vector;
v^-1 is v inverse;
v^T is v transpose;

Assume A is a non singular matrix and v is a vector with comparable dimension. Prove:

(A + vv^T)^-1 = A^-1 - [ (A^-1v)(v^TA^-1) / (1 + v^TA^-1v) ]


Need help please. or at least some tips. Thanks!

What work do you have from your three days of effort?

Multiply out

and



and show that you get both times. Keep in mind that the product is a scalar, and thus commutes with any multiplication.

How does that work for you?

Solved it. Thanks Ackbeet!

You're welcome! Have a good one.