= Solution
The two orthogonal directions $(1,1)^T$ and $(1,-1)^T$ are preserved by this <symmetric matrix>. Direct multiplication, using the <hyperbolic function> identities $\cosh u\pm\sinh u=e^{\pm u}$, gives the <eigenvalue>–<eigenvector> pairs
$$
\boxed{\lambda_+=e^u,\quad \boldsymbol v_+=\binom11;
\qquad \lambda_-=e^{-u},\quad \boldsymbol v_-=\binom1{-1}.}
$$
Each <eigenvector> can be multiplied by any nonzero scalar. For $u\ne0$ these give the two one-dimensional <eigenspaces>. When $u=0$, the <matrix> is the identity: the only <eigenvalue> is $1$, with multiplicity two, and every nonzero <vector> is an <eigenvector>.
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