Determining eigenvectors So let us say we have the following matrix: $$ A = \begin{bmatrix} 1 & -3 \\\ 1 & 5 \\\ \end{bmatrix} $$ I can calculate that the eigenvalues are $4$ and $2$. So now it is needed to calculate $(4I-A)x = 0$ and $(2I-A)x = 0$ in order to get the eigenvectors. The first step is to turn it into an equationsystem which will thereafter be solved to determine the eigenvectors. But how do I go from the expression (for example $(4I-A)x = 0$) to the equationsystem?

The matrix-vector equation

$$\begin{bmatrix}a&b\\\c&d\end{bmatrix}\begin{bmatrix}x\\\y\end{bmatrix}=\begin{bmatrix}A\\\B\end{bmatrix}$$

is equivalent to the equation

$$\begin{bmatrix}ax+by\\\cx+dy\end{bmatrix}=\begin{bmatrix}A\\\B\end{bmatrix}$$