How can I find eigenvectors of the following matrix?
$$ \begin{matrix} 4 & 0 \\ 0 & 1 \\ \end{matrix} $$
Systematic approach would be: 1. Finding eigenvalues (4,1), 2. solve
(A-4I)x = 0 and (A-1I)x = 0.
For some reason I am unable to solve augmented matrices:
For eigenvalue 4:$$ \begin{matrix} 0 & 0 & 0\\ 0 & -3 & 0\\ \end{matrix} $$
and
For eigenvalue 1:$$ \begin{matrix} 3 & 0 & 0\\ 0 & 0 & 0\\ \end{matrix} $$
How can I get eigenvectors (1,0) and (0,1) from those matrices?
For the first eigenvalue, $\lambda_1 = 4$, we form $[A-4I]v_1=0$ and we have a RREF of:
$$\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right)v_1 = 0 \implies v_1 = (1,0)$$
For the second eigenvalue, $\lambda_2 = 1$, we form $[A-1I]v_2=0$ and we have a RREF of:
$$\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right)v_2 = 0 \implies v_2 = (0,1)$$