What is the eigenvalue of matrix from matrix minus all-ones matrix?

Question

Suppose we know the eigenvalues of matrix A, and J is all-ones matrix with all elements are one. Then what are the eigenvalues of A-J?

ps. A is random matrix with element from distribution N(0,s)

Thanks.

Answer 1

Below was my answer to the question as it was originally posed. It has since been modified, and this is no longer a valid answer to the question.

There is no way that this question can be answered with the information given.

For example, $A = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} $ has eigenvalues $1$ and $1$, and $A - J$ has eigenvalues $-1$ and $1$.

If $B = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} $, then $B$ has the same eigenvalues as $A$. However, $B - J$ has the eigenvalues $0$ and $0$.