There is a true/false question that reads:
If $P$ and $Q$ are orthogonal matrices then $P^TQ$ is an orthogonal matrix.
This is the given solution, which I do not understand:
$(P^TQ)^T = Q^TP = Q^{-1}P = Q^{-1}(P^T)^{-1} = (P^TQ)^{-1}$
or here is a picture so it's more readable: https://i.gyazo.com/4f66070bf4f6b5a25a4c0011e0349cc5.png
So I know that if $P$ is orthogonal then $P^T = P^{-1}$
But, right off the bat the proof starts with $(P^TQ)^T$
How can $P^TQ = (P^TQ)^T$ if the transpose of $Q$ doesn't necessarily equal $Q^T$
Can someone explain this to me?
This is an Introduction to Linear Algebra class, so please try to keep the answers as basic as you can.
Best Answer
The book starts with the expression $(P^TQ)^T$. It does not say that $P^TQ = (P^TQ)^T$.
By definition, an orthogonal matrix $X$ must satisfy $X^T = X^{-1}$. So to prove that $P^TQ$ is an orthogonal matrix, we must show that $(P^TQ)^T = (P^TQ)^{-1}$.
That's why the book starts with the expression $(P^TQ)^T$ and manipulates it until they get $(P^TQ)^{-1}$.