$ (x x^T)^{-1}$, efficient matrix inversion for matrix composed as product of a vector with itself?

Question

Given a vector $x$, is there an efficient way of computing $(x x^T)^{-1}$? I mean without first computing the matrix $(x x^T)$ and then applying matrix inversion techniques to it?

Answer 1

As Wim already noted, the matrix is always singular, so inverse never exist for dimensions $\geq 2$ but you can get a pseudoinverse as $$(xx^T)^\dagger = \frac{(xx^T)}{{(x^Tx)}^2}$$The pseudoinverse has the properties that $x$ is mapped back onto itself and all vectors orthogonal to $x$ are mapped to 0. We can confirm this as: $$ (xx^T)^\dagger(xx^T)x = \frac{xx^Txx^Tx}{(x^Tx)^2} = \frac{x(x^Tx)(x^Tx)}{(x^Tx)^2} = x$$