= Solution
For the special-relativistic matrix,
$$
A\binom11=\gamma(1-\beta)\binom11,
\qquad
A\binom1{-1}=\gamma(1+\beta)\binom1{-1}.
$$
Thus the <eigenvalues> and corresponding <eigenvectors> are
$$
\boxed{
\lambda_+=\sqrt{\frac{1-\beta}{1+\beta}},
\quad v_+=\binom11},
\qquad
\boxed{
\lambda_-=\sqrt{\frac{1+\beta}{1-\beta}},
\quad v_-=\binom1{-1}}.
$$
The eigenvector equations are $x=ct$ and $x=-ct$. They are the two <null directions> of the <light cone>, representing right-moving and left-moving light rays. A Lorentz boost preserves each light-ray direction while rescaling its null coordinate.
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