Solution
= Solution
\b[No.] Consider the two orthogonal projections
$$
P=\begin{pmatrix}1&0\\0&0\end{pmatrix},
\qquad
Q=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}.
$$
Their product is
$$
PQ=\frac12\begin{pmatrix}1&1\\0&0\end{pmatrix}.
$$
For $x=(-1,2)^T$,
$$
x^\dagger PQx=-\frac12<0.
$$
\b[Thus $PQ$ is not a <positive semidefinite matrix>. It is also not Hermitian; products of orthogonal projections need not remain orthogonal projections unless the factors commute.]