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Find the $3 \times3$ matrix (all non-zero entries) that has eigenvalues $\lambda = 1, \lambda = 3, \lambda = 5.$

I'm having trouble figuring out how to approach this question. I was thinking of using the characteristic polynomial, but not entirely sure.

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    $\begingroup$ Set $D=\begin{bmatrix}1&0&0\\0&3&0\\0&0&5\end{bmatrix}$ and experiment with different invertible matrices $P$ - producing matrices $P^{-1}DP$ which will all have eigenvalues $1,3,5$ - one of those is bound to have all nonzero entries. In all likelihood, you will stumble upon a suitable matrix $P$ in a few tries if not faster. $\endgroup$
    – user700480
    Feb 27, 2021 at 18:08
  • $\begingroup$ @StinkingBishop: That seems like a great answer to me (i.e. it should be an "Answer"). $\endgroup$
    – Lee Mosher
    Feb 27, 2021 at 18:13

1 Answer 1

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Copying an answer from the comments to take this off the unanswered queue.

Start with $D = \left[ \matrix{1 & 0 & 0 \\0 & 3 & 0 \\0 & 0 & 5} \right]$ and experiment with different invertible $3 \times 3$ matrices $P$, producing matrices $P D P^{-1}$ which will all have eigenvalues $1$, $3$, $5$. In all likelihood you will stumble upon a suitable matrix $P$ in a few tries if not faster.

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