Matrix method for solving linear complex vector functional equations (Q1607837)

Let \(\mathcal V\) be a finite-dimensional complex vector space and let \(f:\mathcal V^n \to \mathcal V\); denote with \(\mathbb{Z}_i\) (\(1\leq i\leq n\)) the vectors in \(\mathcal V\). The main result gives the representation of the general solution of the cyclic non-homogeneous functional equation \[ \sum_{i=1}^n a_if(\mathbb{Z}_i, \mathbb{Z}_{i+1},\cdots,\mathbb{Z}_{i+n-1})=g(\mathbb{Z}_1,\mathbb{Z}_2,\cdots,\mathbb{Z}_n) \quad (\mathbb{Z}_{n+i}=\mathbb{Z}_i) \] where \(a_i\) are complex constants. The method developed is used for solving some other similar equations.