Let $ K $ be a field. We can recursively define matrices as
$ M_{a} = (a)$ for any $ a\in K $ and
$$ M_{a_1, \cdots, a_{2^i}} =
\begin{pmatrix}
M_{a_1, \cdots, a_{2^{i1}}} & M_{a_{2^{i1} +1}, \cdots, a_{2^i}}\\
M_{a_{2^{i1} +1}, \cdots, a_{2^i}} & M_{a_1, \cdots, a_{2^{i1}}}\\
\end{pmatrix}
$$
when $ i>0 $ and $a_j\in K$.
What is the name for the type of matrices?
Let $ a_1, a_2, \cdots, a_{2^n} $ and
$ b_1, b_2, \cdots, b_{2^n} $ be two list of elements in $ K $.
Is there a formula for the eigenvalues of
$$ M_{a_1, a_2, \cdots, a_{2^n}}  \operatorname{diag} ( b_1, b_2, \cdots, b_{2^n})? $$

$\begingroup$ Please add at least one $2\times 2$ example to help us. $\endgroup$– P VanchinathanMar 26, 2014 at 1:00

$\begingroup$ You may regard $M_{a_1,\dots,a_{2^n}}$ as the `addition table' for $n$dimensional vector space over $\mathbb F_2$. $\endgroup$– Ilya BogdanovMar 26, 2014 at 8:30
2 Answers
As for the eigenvalues there is neat trick I learned once: A matrix of the form $\left(\begin{smallmatrix} A & B \\ B & A\end{smallmatrix}\right)$ is conjugate (by $\left(\begin{smallmatrix} I & I \\ I & I \end{smallmatrix}\right)$) to $\left(\begin{smallmatrix} A+B & \\ & AB\end{smallmatrix}\right)$. Therefore there is a recursion for the (multi)set of eigenvalues of your matrix.
EDIT: I just realized that the diagonal matrix you're subtracting messes up this special form of the $M_a$ so that what I described leads only to a solution for the special case of a scalar matrix $b_1=...=b_{2^n}$. Maybe you still can use it anyway.
$\let\eps\varepsilon$Making Johannes' answer more exact, one may see that all matrices $M_{a_1\dots a_{2^n}}$ have a common eigenbasis. Namely, for every $\eps_0,\dots,\eps_{n1}\in\{0,1\}$ one may choose $\mathbf e_i=(e_{i1},\dots,e_{in})$ where $e_{ij}=(1)^{\eps_0p_{j0}+\dots+\eps_{n1}p_{j,n1}}$ and $j1=\overline{p_{j,n1}\dots p_{j0}}$ is the binary expansion of $j1$. The corresponding eigenvalue is $\sum_{j=1}^{2^n}a_j(1)^{\eps_0p_{j0}+\dots+\eps_{n1}p_{j,n1}}$. (The transition matrix is a known Hadamard matrix of order $2^n$.)