= Solution
Choose the unnormalized <eigenvectors>
$$
\mathbf u_+=(1,1)^T,\qquad S\mathbf u_+=3\mathbf u_+,
\qquad
\mathbf u_-=(1,-1)^T,\qquad S\mathbf u_-=-\mathbf u_-.
$$
Their <linear combination> giving the specified <vector> is
$$
\mathbf x=\frac12\mathbf u_++\frac12\mathbf u_-.
$$
Applying the reciprocal <eigenvalues> of the <inverse matrix> therefore yields
$$
\boxed{\mathbf y=\frac16\mathbf u_+-\frac12\mathbf u_-}.
$$
In <Cartesian coordinates>, this is $(-1/3,2/3)^T$, and direct <matrix multiplication> gives $S\mathbf y=(1,0)^T$.
Back to article page
