Let us assume that matrices Q1 and Q2 are orthogonal matrices. I have some questions regarding their characteristics.
Case 1:
In this case, I thought that this relationship would be true. However, it turns out that this is actually false according to the exam answer key.
Since Q1 and Q2 are orthogonal matrices, I know that the l2-norm of Q1x would equal that of x; the l2-norm of Q2y would equal that of y. Thus, would not the l2 norm of Q1x - Q2y be the same as the l2-norm of x - y?
Case 2:
In this case, I found a counterexample where the determinant of Q1 is -1, not 1.
Thus, I thought that this statement was false, but it turned out to be true according to the exam answer key. Why is it the case here?
Any clarification on this is much appreciated.
Thanks!!
You're right (and the answers are wrong) with question 2. With question 1, it would be true that $$\|Q_1x - Q_1 y\| = \|x - y\| = \|Q_2x - Q_2y\|.$$ This doesn't say what happens with $\|Q_1x - Q_2y\|$. See if you can find a linear isometry $Q_1$ that maps $(1, 0)$ to $(1, 0)$, and another linear isometry $Q_2$ that maps $(0, 1)$ to $(1, 0)$. Then take $x = (1, 0)$ and $y = (0, 1)$.