1

Given scalars $\alpha_1, \alpha_2, \dots, \alpha_\ell \neq 0$ and unitary matrices $U_1, U_2, \dots, U_\ell$, let

$$ A := \sum_{k=1}^{\ell} \alpha_k U_k $$

where the $\ell$ coefficients $\alpha_1, \alpha_2, \dots, \alpha_\ell \neq 0$ are chosen such that matrix $A$ is invertible. Are there theorems that can help one calculate $A^{-1}$?

I know that for the case where $\ell = 2$ there is the nice Ken Miller (1981) paper, but it isn't obvious to me that I can extend those results to the problem where $\ell > 2$. On Math SE, there is also the nice Inverse of the sum of matrices, which discusses the $\ell = 2$ problem, but I started a new question because my case is quite a bit different.

Rodrigo de Azevedo
  • 1
thespaceman
  • 111
  • 1
    I would describe $A$ not as a sum of unitary matrices but as a linear combination of them. – hardmath Apr 17 '21 at 23:11
  • 3
    It may be possible to optimize for small numbers, but do bear in mind that any operator on any Hilbert space is a linear combination of unitaries (see e.g. Kadison and Pedersen, "Means and Convex Combinations of Unitary Operators," which is nowhere near an optimal result of its type) – leslie townes Apr 18 '21 at 17:31

0 Answers0