The restriction operation (maps a $2n$ vector into an $n$ vector) is defined as:
$$R = \frac{1}{4}\begin{bmatrix} 1& 2& 1 \\ &&1 & 2 &1\\ && \vdots\\ &&&& 1& 2& 1 \end{bmatrix}$$
The interpolation operator (maps an n vector into a 2n vector) is:
$$P = \frac{1}{2}\begin{bmatrix} 1 \\ 2 \\ 1 & 1\\ &2\\ &1 &1\\ &&\vdots\\ &&&1&1\\ && &&2\\ &&&&1\\ \end{bmatrix}$$
These are rectangular matrices and thus do not have egienvalues, however if you apply them to a matrix, e.g. if $B^{n \times n} = PA^{n/2 \times n/2}R$ the resulting matrix does have eigenvalues. Is it possible to define the eigenvalues of $B$ in terms of those of $A$ for both operators?