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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.

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Hint:

$$(A-sI)x_1 = Ax_1 - sIx_1 = a_1 x_1 - sx_1 = (a_1-s)x_1$$

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  • $\begingroup$ Does $a_1$ in your hint represent an entry in the $A$ matrix or the eigenvalue $a_1$? $\endgroup$ – Tightrope Oct 28 '19 at 20:07
  • $\begingroup$ If $a_1$ is an eigenvalue of $A$ then $(a_1 - s)$ is an eigenvalue of $A - sI$. Am I right? $\endgroup$ – Tightrope Oct 28 '19 at 20:09
  • $\begingroup$ The eigenvalue, of course. After all, that is how we were allowed to make that simplification $\endgroup$ – JMoravitz Oct 28 '19 at 20:09
  • $\begingroup$ And $(a_1 - s)x_1$ the corresponding eigenvector? $\endgroup$ – Tightrope Oct 28 '19 at 20:12
  • $\begingroup$ Sure... so long as it isn't zero, but more importantly x1 is $\endgroup$ – JMoravitz Oct 28 '19 at 20:16

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