Characteristic polynomial of a block matrix

Question

Let $A$ be an $n\times n$ symmetric matrix whose diagonal are is covered by zero blocks (square, but not of a fixed size) and all other entries are $1$ (one).

How can I find its Characteristic Polynomial and eigenvalues?

Answer 1

I consider a special case only leaving a (straightforward) generalization to you.

Let the number of blocks be two, i.e. we have the $(k+m)\times(k+m)$ matrix $$ \left[\matrix{0_{k\times k} & 1_{k\times m}\\1_{m\times k} & 0_{m\times m}}\right] $$ where $0_{k\times k}$ and $1_{k\times m}$ denote matrices of all zeros, resp. ones, of the subindexed size.

It is clear that the rank of this matrix is two since there are only two linearly independent rows. It gives you directly that all eigenvalues are zero except maybe two. Let's try to find those two by definition $$ \left[\matrix{0_{k\times k} & 1_{k\times m}\\1_{m\times k} & 0_{m\times m}}\right]\left[\matrix{a\cdot 1_k \\b\cdot 1_m}\right]= \left[\matrix{mb\cdot 1_k \\ka\cdot 1_m}\right]=\lambda\left[\matrix{a\cdot 1_k \\b\cdot 1_m}\right]. $$ Thus $$ \left\{ \begin{array}{l} \lambda a=mb,\\ \lambda b=ka \end{array} \right.\qquad\Rightarrow\lambda^2=mk,\quad \Rightarrow\lambda=\pm\sqrt{mk}. $$