Does every semi-orthogonal matrix reduce norms of vectors?

Question

Matrix $Q$ is semi-orthogonal if $Q^TQ = I$ or $QQ^T = I$. If the first equation is true, then $Q$ is isometric. But what if $QQ^T = I$? Does $Q$ have any special properties regarding vector norms? Specifically, does it maybe always reduce norms, i. e. is it true that $||x||_2 \geq ||Qx||_2$?

Answer 1

Yes, your statement is true. To see that this is the case, note that $$ \|Qx\|_2^2 = (Qx)^T(Qx) = x^T(Q^TQ)x. $$ Now, note that $Q^TQ$ is symmetric and that $(Q^TQ)^2 = Q^TQ$. In other words, $Q^TQ$ is an orthogonal projection matrix and the statement follows.

Alternatively, the fact that $x^T(Q^TQ)x \leq x^Tx$ can be seen as a consequence of the Rayleigh-Ritz theorem, where we note that the fact that $QQ^T$ is symmetric with $(QQ^T)^2 = (QQ^T)$ means that the eigenvalues of $QQ^T$ are equal to $0$ or $1$, so that $x^T(Q^TQ)x \leq 1 \cdot x^Tx$.