# Homework Help: Inverse of outer-product

1. Jul 8, 2010

Hi All,

This is not a homework or coursework question. Yet I have a curiosity.

If a matrix $$A$$ is an outer-product of a vector $$v$$ as : $$A = v v^{\top}$$

Then can $$A^{-1}$$, inverse of $$A$$, be also expressed as an outer-product of some other vector?

Please point me how to approach the question, how to find the answer or how to say it is possible or not.

Thanks.

2. Jul 8, 2010

### Hurkyl

Staff Emeritus
If A is 0x0 or 1x1, inverting it is easy. Otherwise...

Exercise: If v is nonzero, prove that A has rank 1.

You can do better:

Exercise: A matrix has rank 1 if and only if it is the outer product of two nonzero vectors.

3. Jul 8, 2010

oops!! Thanks.