Why $L$ is the eigenspace of $L_A$? $A=\frac{1}{\sqrt{5}}\begin{pmatrix}  1&2\\2&-1 \end{pmatrix}$
Let $L_A$ be a reflection of $R^2$ about a line $L$ through the origin.
Then $L$ is the one dimensional eigenspace of $L_A$ corresponding to the eigenvalue 1.
I know the eigenvalues are 1,-1. But why it should be 1?

Because having an eigenvalue 1 in a direction means that vectors in that direction are not changed, so since your reflection has a unique fixed line, it must be in the eigendirection corresponding to 1.
