I am searching for a keyword that would open the door to further readings. A polynomial matrix is an object of the form
$\mathcal{A}(x) = A_0 + A_1x + A_2 x^2 + \ldots + A_n x^n$
where the $\lbrace A_i \rbrace$ are all $m\times k$ matrices.
What is the name of the following object?
$\mathscr{A}(x) = A_0 + A_1 \phi_1(x) + A_2 \phi_2(x) + \ldots + A_n \phi_n(x)$
where the $\lbrace\phi_i(x)\rbrace$ are (at least) linearly independent scalar functions on the variable $x$ (they could be orthogonal, as in Fourier basis).
Thanks!
UPDATE: So far I have found some characterization of general analytic matrix functions in Chapter S6 of Matrix Polynomials by I. Gohberg, P. Lancaster, L. Rodman. Although analytic matrix function does not describe the structure given above, I guess one can go with that name.