Eigenvalues and inverse of matrix which has matrix entries Suppose I have a $2\times2$ matrix which contains $2\times2$ matrices as entries, is there a clean way to find the inverse of such object and compute its eigenvalues/vectors?
 A: Let $R:=\mathbb R^{2,2}$ the ring of $2x2$ matrices. This ring is not commutative, which complicates everything.
Take a matrix $A=\pmatrix{a_{11}& a_{12}\\a_{21}&a_{22}} \in R^{2,2}$. 
Assume that:


*

*$a_{11}a_{22}= a_{22}a_{11}$

*$a_{12}a_{21} = a_{21}a_{12}$

*$\det(A):= a_{11}a_{22} - a_{12}a_{21}$ is invertible


then $A$ is invertible and
$$
A^{-1} = (\det A)^{-1} \pmatrix{a_{22}& -a_{12}\\-a_{21}&a_{11}}.
$$
This can be checked similarly to the case of a $\mathbb R^{2,2}$ matrix.
This approach works for the permutation matrix
$$
P=\left( \begin{array}{cc|cc}
  1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 1 \\
\hline
  0 & 0 & 1 & 0 \\
  0 & 1 & 0 & 0 \\
 \end{array}\right)
$$
as the diagonal and off-diagonal entries commute. and $\det A=\pmatrix{1&0\\0&-1}$ is invertible.
A: Well, the inverse of a 2x2 block matrix
$$A = \left(\begin{array}{cc} A_{11} & A_{12}\\ A_{21}&A_{22}
\end{array}\right)$$
is (just like a normal 2x2 matrix)
$$A^{-1} = \frac{1}{\det A} \left(\begin{array}{cc} A_{22} & -A_{12}\\ -A_{21}&A_{11}
\end{array}\right)$$
A: A $2 \times 2$ matrix whose entries are $2 \times 2$ matrices themselves is exactly the same thing as a $4 \times 4$ matrix, so you can use any of the usual matrix techniques.
A: If the matrices $A_{11},\;A_{12},\;A_{21},\;A_{22}$ do not have any special properties, then there is nothing that makes $A$ special compared to any other $4\times 4$ matrix. Therefore, as quarague already mentioned, you can use any of the usual matrix techniques.
If the blocks have special properties, then it may be possible to simplify the inversion and eigenvalue/eigenvector calculation.
One thing that comes to my mind is the commutative property. If $A_{11},\;A_{12},\;A_{21},\;A_{22}$ all commute, you get
$$
\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}^{-1}
=
\begin{pmatrix} (A_{11}A_{22}-A_{12}A_{21})^{-1} & 0 \\
0 & (A_{11}A_{22}-A_{12}A_{21})^{-1} \end{pmatrix}
\begin{pmatrix} A_{22} & -A_{12} \\ -A_{21} & A_{11} \end{pmatrix}
$$
