Solution
= Solution
A real square <matrix> $Q$ is an <orthogonal matrix> when $Q^TQ=I$, equivalently its columns form an <orthonormal basis>. Here $Q^T$ is the <transpose> and $I$ the <identity matrix>. For every <vector> $x$,
$$
\|Qx\|_2^2=(Qx)^T(Qx)=x^TQ^TQx=x^Tx=\|x\|_2^2.
$$
Taking nonnegative <square roots> proves
$$
\boxed{\|Qx\|_2=\|x\|_2.}
$$
Thus an <orthogonal matrix> preserves the <Euclidean norm>.