We're all familiar with the many computational methods to solve the standard linear system
$$ Ax=b. $$However, I'm curious if there are any "standard" computational methods for solving a more general (finite dimensional) linear system of the form
$$ LA=B, $$ where, say, $A$ is an $m_1\times n_1$ matrix, $B$ is an $m_2\times n_2$ matrix, and $L$ is a linear operator taking $m_1\times n_1$ matrices to $m_2\times n_2$ matrices, which does not involve vectorizing the matrices, i.e. converting everything to the standard $Ax=b$ form.
The reason I ask is I need to solve the following equation for $u$:
$$ (R^*R+\lambda I)u=f $$ where $R$ is the 2d Radon transform, $R^*$ its adjoint, and both $u$ and $f$ are 2d arrays (images). It is possible to vectorize this equation, but it's a pain, especially if we go to 3D.
More generally, what about $nD$ arrays? For instance, solving $LA=B$ where $A$ and $B$ are 3D arrays (I'll need to do this with Radon transform at some point as well).
Thanks ahead, and feel free to ship me off to another StackExchange if you feel the need.