I have a question to ask down below, that I have been having some trouble with and would like some help and clarification on.
Suppose A is an $n \times n$ matrix with (not necessarily distinct) eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$. Can it be shown that:
(a) The sum of the main diagonal entries of A, called the trace of A, equals the sum of the eigenvalues of A.
(b) A $- ~ k$ I has the eigenvalues $\lambda_{1}-k, \lambda_{2}-k, \ldots, \lambda_{n}-k$ and the same eigenvectors as A.
Thank You very much.