Isometries (reflection matrix) and eigenvectors

Question

Suppose I have a matrix $$F= \begin{pmatrix} 0.6&0.8\\ 0.8&-0.6 \end{pmatrix} $$ which is clearly a reflection matrix.

Furthermore, I have found $\lambda_1 = 1,\vec{v}_1=\begin{pmatrix} 2\\ 1 \end{pmatrix}$ and $\lambda_2=-1, \vec{v}_2=\begin{pmatrix} 1\\ -2 \end{pmatrix}$, how do I exploit the fact that the eigenvectors associate with these two eigenvalues form a basis to determine the effect of $F$ on a vector $\vec{w}= \begin{pmatrix}3\\ -1 \end{pmatrix}$ which will give $F\vec{w}=\begin{pmatrix} 1\\ 3 \end{pmatrix}$.

Answer 1

$w = v_1 + v_2$

$Fw = Fv_1 + Fv_2 = \lambda_1v_1 + \lambda_2 v_2 = v_1 - v_2$

Answer 2

Your eigenvalues indicate that one eigenvector is preserved by $F$, and the other is reversed in direction. The one preserved lies on the axis of symmetry through which $F$ reflects.

Thus, you want to write your $\vec{w}$ as a linear combination of these two eigenvectors. And then apply $F$.