Can "eigenvalues" in eigenfunction expansion be non-scalar?

Question

This question is a bit nebulous but I don't have a particular example in mind... In general under certain assumptions, one can use eigenfunction expansion to represent an operator. For instance, for an operator $\mathcal{L}$ $$\mathcal{L} \psi_i = \lambda_i \cdot \psi_i \; ,$$ where $i = 1, \cdots, \infty$ for an orthonormal system $\psi_i$ dense in some space. Can $\lambda_i$ be a vector or matrix in such expansion, hence each $\psi_i$ is a vector or a matrix as well? I don't see a reason why it cannot be, if the operator $\mathcal{L}$ can be a map from $\mathbb{R}^k$ to $\mathbb{R}^k$. But if such scenario does exist, could you please point me to some reference using such expansion technique? If such scenario cannot exist, could you explain why? Thanks in advance!