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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.

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    For (a), see http://en.wikipedia.org/wiki/Trace_%28linear_algebra%29 . For (b), use the equation that defines eigenvalues and eigenvectors.2011-03-15

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