Is matrix $Svv^\top S$ positive semidefinite, given $S$ is symmetric positive semidefinite?

Question

Is matrix $Svv^\top S$ positive semidefinite, given $S$ is symmetric positive semidefinite? Where $v\in\mathbb{R}^n$, $S\in \mathbb{R}^{n\times n}$

Answer 1

Yes, if your definition of positive semidefinite includes that $S$ is symmetric. Then $$ Svv^\top S=Svv^\top S^\top=(Sv)(Sv)^\top. $$ $$ A\triangleq (Sv)^\top\quad\Rightarrow\quad (Sv)(Sv)^\top=A^\top A $$ Any matrix of the form $A^TA$ is positive-semidefinite: because if $A$ is $m\times n$, then for any $x$ of size $n\times 1$, $$ x^T(A^TA)x=(Ax)^TAx=y^Ty=\sum_{j=1}^my_j^2\geq0, $$ where $y=Ax$.