How to find a real matrix with complex eigenvalues,

Question

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$.

I would like hints only.

So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which gives me a set of two equations:

1) $(x_1-2-3i)(y_2-2-3i)-x_2y_1 = 0$

2) $(x_1-2+3i)(y_2-2+3i)-x_2y_1 = 0$,

where $x_1$ is the first entry of $A$, $x_2$ is the $a_{1,2}$ entry of $A$, and so on.

I'm getting nowhere with this method, though...

Thanks,

Answer 1

Hint: $Tr(A) = A_{11} + A_{22} = 4$ (sum of eigenvalues)

Another Hint: $Det(A) = A_{11} * A_{22} - A_{12} * A_{21}= 13$ (product of eigenvalues)

Answer 2

Hint: start by finding the characteristic polynomial of $A$, and try working from there.

Answer 3

Hint: If $a$ and $b$ are real numbers, what are the eigenvalues of $\begin{bmatrix}a & -b \\ b & a\end{bmatrix}$?

Alternatively, notice that $\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$ (the matrix which represents a rotation by $180^{\circ}$) has eigenvalues $\pm i$. Can you manipulate this matrix into one which has eigenvalues $2 \pm 3i$?

Answer 4

Hint: What is the trace and determinant of the matrix?