Show that $PQ - I_n$ is singular when $Q - I_n$ is not Let $Q$ be an $n \times n$ real orthogonal matrix and $u \in \mathbf{R}^n$ be a unit vector. Let $P = I_n - 2uu^\top$ be the Householder reflection. Show that if $Q - I_n$ is not singular then $PQ - I_n$ is singular. 
 A: When $n$ is odd, $Q$ has a fixed point whenever it has determinant $1$. When $n$ is even, $Q$ has a fixed point whenever it has determinant $-1$. These facts can be seen by considering the eigenvalues of $Q$: there are always $n$ of them, they are closed (as a multiset) under conjugation, and they must multiply to the determinant. It follows that, when $Q$ does not have a fixed point, multiplying it by any negative-determinant orthogonal matrix produces a matrix with a fixed point.
A: This answer isn't as neat as Christopher Gadzinski's, but it gives an explicit non-trivial solution to $(PQ-I)v=0$. Note that
\begin{aligned}
(I-Q)^{-1}+(I-Q^T)^{-1}
&=(I-Q)^{-1}\left[(I-Q^T)+(I-Q)\right](I-Q^T)^{-1}\\
&=(I-Q)^{-1}\left[(I-Q)(I-Q^T)\right](I-Q^T)^{-1}=I.
\end{aligned}
Since the LHS is the double of the symmetric part of $(I-Q)^{-1}$, the above implies that $2(I-Q)^{-1}=I+K$ for some skew-symmetric matrix $K$. Hence $1-2u^T(I-Q)^{-1}u=0$,
\begin{aligned}
-P(PQ-I)(I-Q)^{-1}u
&=(P-Q)(I-Q)^{-1}u\\
&=(I-Q-2uu^T)(I-Q)^{-1}u\\
&=\left[1-2u^T(I-Q)^{-1}u\right]u=0
\end{aligned}
and $PQ-I$ is singular.
