Let $A$ be a $3*3$ matrix with eigenvalues $a_1$, $a_2$ and $a_3$ and corresponding eigenvectors $x_1$, $x_2$ and $x_3$. Let $s$ be a given number. Suppose that $a_1$ is the eigenvalue nearest to $s$ and $a_2$ is the eigenvalue furthest from $s$, and let $I$ denote the $3*3$ identity matrix.
i) What are the eigenvalues and the corresponding eigenvectors of the matrix $A – sI$?
ii) What is the dominant eigenvalue of the matrix $A – sI$?
iii)What is the eigenvalue of the matrix $A – sI$ with the smallest absolute value?
Because this is more of a theoretical than a practical question, I have difficulty making sense of it. I know how to work out the dominant eigenvalue and the smallest eigenvalue using the power method on an actual matrix with entries, but it seems this question wants the answer in mathematical notation.
Best Answer
Hint:
$$(A-sI)x_1 = Ax_1 - sIx_1 = a_1 x_1 - sx_1 = (a_1-s)x_1$$