Relationship between two matrices

[trenekas]

Hello. I need some help with one question about relationship of two matrices.
The task:
Suppose that I is identity matrix, u - is vector, u' is transposed vector, α - real number. It can be prove that inverse matrix of I+α*u*u' has similar form I+x*u*u'. The task is to find x.

I tried to calculate inverse of this matrix I+α*u*u', but first of all when i don't know dimension it is very difficult to calculate inverse matrix. I tried to take example when u is from R^2, but even then i can't calculate x.

Any help would be appreciate. Any hints or something else.

 

[trenekas]

I think here is need to use Woodbery formula, http://en.wikipedia.org/wiki/Woodbury_matrix_identity
but i can't get the answer. :/

 

[Matterwave]

Instead of trying to calculate the inverse, why not assume the inverse is like I+x*u*u' and see what happens when you take the product?

$$(I+\alpha uu^T)(I+xuu^T)=?$$

 

[trenekas]

ok i will try. thanks for help, Matterwawe!

 

[trenekas]

$$(I+\alpha uu^T)(I+xuu^T)=I$$
$$(I+I*xuu^T+I*\alpha uu^T+\alpha uu^T*xuu^T=I$$
So after that
$$I*xuu^T+I*\alpha uu^T+\alpha uu^T*xuu^T=0$$
$$xuu^T+\alpha uu^T+\alpha uu^T*xuu^T=0$$
$$(x+\alpha) uu^T+\alpha*x (uu^T*uu^T)=0$$
$$(x+\alpha) uu^T=-\alpha*x (uu^T*uu^T)$$

And what's next? How can i express x?

$$(x+\alpha)/-\alpha*x =(uu^T)^{-1}(uu^T*uu^T)$$
$$-1/\alpha-1/x=(uu^T)^{-1}(uu^T*uu^T)$$
$$-1/x=(uu^T)^{-1}(uu^T*uu^T)+1/\alpha$$

 

[Regensburger]

From equation
$$xuu^T+\alpha uu^T+\alpha uu^T*xuu^T=0$$
follows with
$$uu^T*xuu^T = xu(u^Tu)u^T = x (u^Tu) uu^T$$ (u^Tu is a scalar)
the solution of your question:
$$x = -\alpha / (1+\alpha u^Tu)$$