TITLE: eigenvalues of certain block matrices
QUESTION [13 upvotes]: This question inquired about the determinant of this matrix:
$$
\begin{bmatrix}
  -\lambda &1  &0  &1  &0  &1  \\
   1& -\lambda &1  &0  &1  &0  \\
   0&  1& -\lambda &1  &0  &1  \\
   1&  0&  1& -\lambda &1  &0  \\
   0&  1&  0&  1& -\lambda &1  \\
   1&  0&  1&  0&1 & -\lambda
\end{bmatrix}
$$
and of other matrices in a sequence to which it belongs.  In a comment I mentioned that if we permute the indices 1, 2, 3, 4, 5, 6 to put the odd ones first and then the even ones, thus 1, 3, 5, 2, 4, 6, then we get this:
$$
\begin{bmatrix}
-\lambda & 0 & 0 & 1 & 1 & 1 \\
0 & -\lambda & 0 & 1 & 1 & 1 \\
0 & 0 & -\lambda & 1 & 1 & 1 \\
1 & 1 & 1 & -\lambda & 0 & 0 \\
1 & 1 & 1 & 0 & -\lambda & 0 \\
1 & 1 & 1 & 0 & 0 & -\lambda
\end{bmatrix}
$$
So this is of the form
$$
\begin{bmatrix}
A & B \\ B & A
\end{bmatrix}
$$
where $A$ and $B$ are symmetric matrices whose characteristic polynomials and eigenvalues are easily found, even if we consider not this one case of $6\times 6$ matrices, but arbitrarily large matrices following the same pattern.
Are there simple formulas for determinants, characteristic polynomials, and eigenvalues for matrices of this latter kind?
I thought of the Haynesworth inertia additivity formula because I only vaguely remembered what it said.  But apparently it only counts positive, negative, and zero eigenvalues.

REPLY [2 votes]: Your $2n\times 2n$ matrix $M$ acts on the vector space $V=\mathbb C^n\oplus\mathbb C^n$. Now if $W_1=\{(v,v):v\in\mathbb C^n\}$ and $W_2=\{(v,-v):v\in\mathbb C^n\}$, then we also have $V=W_1\oplus W_2$. Moreover, both $W_1$ and $W_2$ are invariant under $M$, so to find the eigenvalues/eigenvectors/characteristic polynomial/etc, it is enough to do it for those restrictions: they are $A+B$ and $A-B$.
This way you obtain, for example, the facts mentioned in Sunni's answer immediately.