If $Y\sim N(\mu,\mathtt{V})$ and $\mathtt{A}$ is symmetric then prove that \begin{equation} Y^\top\mathtt{A}Y\sim\chi_{r,\mu^\top A\mu}^2\text{ if and only if }\mathtt{AV}\text{ is idempotent of rank }r\text{. } \end{equation}
$\chi_{r,\mu^\top\mathtt{A}\mu}^2$ represents the noncentral chi squared distribution with $r$ degrees of freedom and noncentrality $\mu^\top\mathtt{A}\mu$.
What I have tried
I know that $Y=\mathtt{B}Z+\mu$, where $Z\sim N(0,\mathtt{I})$ and $\mathtt{B}$ is a regular matrix such that $\mathtt{BB}^\top=\mathtt{V}$, then I concluded that \begin{equation} Y^\top \mathtt{A}Y=Z^\top\mathtt{B}^\top\mathtt{AV}\mathtt{B}^{-\top}Z+2\mu^\top\mathtt{AB}Z+\mu^\top\mathtt{A}\mu\text{, } \end{equation} I this point I tried to expand some terms using the idempotency of $\mathtt{AV}$, but it seems to be a dead end, also, I can't figure out how to use the rank hypothesis.