Is the Inverse of an Outer-Product Matrix Also an Outer-Product?

[omaradib]

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.

 

[Hurkyl]

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

Then can $$A^{-1}$$

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.

 

[omaradib]

oops! Thanks.