Low-rank matrices (and tensors) appear often in applications, and this can

be exploited in several ways to treat large scale problems that would be otherwise

unfeasible. It has been known since a long time that in some applications, such

as the solution of the Sylvester equation AX + XB + C = 0, working with

low-rank C often leads to a solution X which is full rank, but can be efficiently

approximated by a low-rank matrix. This can be used, among other things, in

solving PDEs defined by separable operators, such as the 2D Laplace operator

on rectangular domains.

We briefly review some of these results, and the connection with some ra-

tional approximation problems. We show that these structures appear in a

multitude of interesting cases. For instance, the same properties can be found

when modifying the coefficients of linear matrix equation by performing low-

rank updates to their coefficients, and the theory can be extended to more

general matrices which have off-diagonal blocks of low-rank.

Time permitting, we discuss some recent developments that allow to prove

the existence of a low-rank structures in the action of functions of matrices that

have a Kronecker sum structure, such as x = f (A⊗I + B^T⊗I)v, for particularly

structured vector v. An efficient algorithm for the approximation of the vector

x, reshaped in matrix form, is given. This founds applications in extending

the ideas used for the Laplace operator and 2D PDEs to more general nonlocal

operators, such as the fractional Laplacian.