Zero quadratic form of a positive semidefinite matrix
= Zero quadratic form of a positive semidefinite matrix
If $P$ is a real <positive semidefinite matrix>, then $x^TPx=0$ if and only if $Px=0$. Indeed $x^TPx=\|\sqrt P\,x\|^2$ by the <principal square root of a positive semidefinite matrix>, so a zero <quadratic form> puts $x$ in the <kernel> of $\sqrt P$ and hence of $P$.