# randomized approximate matrix inverse or adjoint of a square matrix

I have been reading about some random matrix theory, JL, and related topics and am wondering if there are any methods to calculate an approximate inverse of a SPD matrix $\mathbf{A}$, or possibly even the adjoint of that matrix. I had a look around the literature and came up with not a single solution - I may be searching the wrong keywords. My intuition is that if there is something out there it might be related to approximate solutions that rely on approximate calculations of log determinants. I couldn't find any relevant literature.

An initial thought was that through the relation $\mathbf{A}^{-1}=\frac{adj(\mathbf{A})}{det(\mathbf{A})}$, if one knows $\hat{det}(\mathbf{A})$, and $\hat{adj}\mathbf{A})$ it might be possible to then calculate $\hat{\mathbf{A}}^{-1}$.

I know there is a reasonably large body of research into approximate log determinants, e.g. here, and was wondering if there is anything akin to this using random projections or iterative methods for recovering an approximate inverse $\mathbf{A}^{-1}$

Now, according to this page there are many names and notations for the adjoint: "The conjugate transpose is also known as the adjoint matrix, adjugate matrix, Hermitian adjoint, or Hermitian transpose (Strang 1988, p. 221)." I could not however find anything fundamental regarding approximations of any of these alternative terms.

I would be extremely grateful if anyone is familiar with the literature could point me to any relevant research

thanks!

p.s. I am familiar with methods such as the sherman-morrison update, cholesky and related factorisations and am more interested in applications of random matrix theory for the above problem

• You've misunderstood Strang's quote- the adjoint that's related to $A^{-1}$ is the transpose of the matrix of cofactors of $A$. The same word is more commonly used to refer to the transpose of a matrix. – Brian Borchers Jul 10 '18 at 3:57
• Although I'm familiar with fast randomized algorithms for approximate solution of least squares problems, I've not seen any work on randomized algorithms for the inverse of a matrix. – Brian Borchers Jul 10 '18 at 3:58
• @BrianBorchers thanks for the input Brian; I'll keep a look out – undercurrent Jul 25 '18 at 0:40