MGF of quadratic form $\mathbf{y}^\intercal \mathbf{A} \mathbf{y}$ where $y\sim N_p(\mathbf{\mu},\mathbf{\Sigma})$

Question

Theorem 5.2b of Linear Models in Statistics by Rencher and Schaalje is

If $\mathbf{y}$ is $N_p(\mathbf{\mu},\mathbf{\Sigma})$, then the moment generating function of $\mathbf{y}^\intercal\mathbf{A}\mathbf{y}$ is \begin{equation} M_{\mathbf{y}^\intercal\mathbf{A}\mathbf{y}}(t)=|\mathbf{I}-2t\mathbf{A}\mathbf{\Sigma}|^{-1/2}\exp(-\mathbf{\mu}^\intercal[\mathbf{I}-(\mathbf{I}-2t\mathbf{A}\mathbf{\Sigma})^{-1}]\mathbf{\Sigma}^{-1} \mathbf{\mu}/2) \end{equation}

I do not understand two parts of the proof:

1. The text asserts that "for $t$ sufficiently close to $0$, $\mathbf{I}-2t\mathbf{A\Sigma}$ is nonsingular." How can I prove this?
2. At one point in the proof, the random vectors $\mathbf{\theta}^\intercal=\mathbf{\mu}^\intercal(\mathbf{I}-2t\mathbf{A\Sigma})^{-1}$ and $\mathbf{V}^{-1} = (\mathbf{I} - 2t\mathbf{A\Sigma})\mathbf{\Sigma}^{-1}$ are defined to use the product $\mathbf{\theta}^\intercal \mathbf{V}^{-1}=\mathbf{\mu}^\intercal\mathbf{\Sigma}^{-1}$. It is then stated that $\mathbf{V}$ (and $\mathbf{\Sigma}$) is symmetric, so $\mathbf{V}^{-1}\mathbf{\theta}=\mathbf{\Sigma}^{-1}\mathbf{\mu}$.

I understand how to get $\mathbf{V}^{-1}\mathbf{\theta}=\mathbf{\Sigma}^{-1}\mathbf{\mu}$ from $\mathbf{V}$ and $\mathbf{\Sigma}$ being symmetric but how do I prove that $\mathbf{V}$ is symmetric?

Answer 1

I assume that $A$ is symmetric. Then, the second claim follows by taking the transpose from $V^{-1}$, and proving that $V^{-1}$ is symmetric. Then, note that given an invertible matrix is symmetric then the matrix itself is also symmetric.