Solution
= Solution
For the specified coordinate vectors,
$$
L=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad M=\begin{pmatrix}1&0\\-1&1\end{pmatrix}.
$$
Choose the <orthogonal matrix> representing a quarter-turn,
$$
\boxed{S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.}
$$
Then $S^TS=I$ and direct <matrix multiplication> gives $S^TLS=\begin{pmatrix}1&0\\-1&1\end{pmatrix}=M$. Thus these two <shear mappings> are orthogonally similar.