If $A$ is symmetric and $Y\sim\mathcal N(0,V)$, how can I show that $Y'AY\sim\sum_{i=1}^{t}(c_i * \chi^2(1))$ with 1 degree of freedom), where $c_i$ can be any scalar?
I multiplied out the canonical case where length($Y$) is two, and got $$ \begin{aligned} Y'AY &= [Y_1 Y_2] \left[\begin{array}{cc} a_{11} & a_{12} \\ a_{12} & a_{22} \\ \end{array}\right] \left[\begin{array}{cc} Y_1 \\ Y_2 \end{array}\right] \\ &= [Y_1 Y_2] [a_{11}Y_1 + a_{12}Y_2 ; a_{12}Y_1 + a_{22}Y_2] \\ &= a_{11} * Y_2^2 + 2a_{12}Y_1Y_2 + a_{22}Y_2^2 \end{aligned} $$ obviously the first and last term are chi-squared of degree 1, but the cross term? As $n$ increases (where the above example is $n = 2$), the cross terms increase as well.