[Math] Find all eigenvalues and corresponding eigenvectors for the matrix

Question

Find all eigenvalues and corresponding eigenvectors for the matrix:
$$
\left(\begin{array}{cr}
0&-1 \\
2&3
\end{array}\right)
$$
Not looking for a answer, but I don't know what an "eigenvalue" is or how to find them. Can anyone help me here with a jump off point to get started?

Answer 1

To compute the eigenvalues solve $det \begin{pmatrix}0-\lambda&-1\\2&3-\lambda\end{pmatrix}=0$. You will get $\lambda=1,2$. These are the two eigenvalues. For each of these you need to solve the system of equations given by $\begin{pmatrix}0-\lambda&-1\\2&3-\lambda\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$. The non-zero solutions are the eigenvectors.

For example with $\lambda=1$ we have the system given by $\begin{pmatrix}-1&-1\\2&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ which has a non zero solution $(1,-1)$.

With $\lambda=2$ we have the system given by $\begin{pmatrix}-2&-1\\2&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ which has a non zero solution $(1,-2)$.

In this way you have found both eigenvalues and a corresponding eigenvector for both.