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

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.

• 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.
– user700480
Feb 27, 2021 at 18:08
• @StinkingBishop: That seems like a great answer to me (i.e. it should be an "Answer"). Feb 27, 2021 at 18:13

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.