A matrix $M$ is positive semidefinite if and only if $y^T M y \geq 0$ for all possible $y$. We can also show that $M \succeq 0$ if and only if $[\bar{y}^T,1 ]M[\bar{y}^T,1]^T \geq 0$ for all possible $\bar{y}$. That is we fix the last element of $y$ to be 1. This can be proved by the continuity of $ y \rightarrow y^T M y$.
My question is: is this well-known? If so, is there a reference for this?