# Finding a matrix that has complex Eigenvalues

I have an assignment where I need to create 2x2 matrices for each of the following Eigenvalue pairs.

i) eigenvalues are 2 and -3
ii) eigenvalues are 2+i and 2-i
iii) eigenvalues are 1/2+2i and 1/2-2i


I got the first one just by putting a 0 in one corner and 2 -3 along the diagonal, but I really have no clue how to get an complex number for an eigenvalue. Please someone help me understand!

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Note that 1/2+2i and 1/2-2i are ambiguous. Please use either parentheses or horizontal fraction bars to clarify whether you mean $\frac{1}{2+2i}$ and $\frac{1}{2-2i}$, or $\frac{1}{2}+2i$ and $\frac{1}{2}-2i$. – Jonas Meyer Dec 8 '11 at 5:13
Do you have a restriction on the entries? If not, why not take a matrix that has $2+i$ and $2-i$ in the diagonal, and a zero in one corner? If the entries must be real, note that the characteristic polynomial needs to be $(t-(2+i))(t-(2-i)) = t^2 - 4t + 5$, so fill in your matrix with $a$, $b$, $c$, $d$, compute the characteristic polynomial in terms of $a$, $b$, $c$, and $d$, and find some real solutions to the corresponding equations. – Arturo Magidin Dec 8 '11 at 5:15
yes they have to be real entries. I think this will help tho – NSjonas Dec 8 '11 at 5:18

It's pretty easy to generate a $2\times 2$ matrix with real entries and conjugate eigenvalues. If you have
$$\begin{pmatrix}a&c\\d&a\end{pmatrix}$$
then the eigenvalues of this matrix are $a\pm\sqrt{cd}$.