Second Derivative of a matrix Pardon me for not knowing LateX representation,
I have following function, where $\mu$ and $\Sigma$ are both Matrices.
$$ h = \mu^T \Sigma \mu $$
which is a function of $\alpha$, such that its derivative can be written as
$$ \def\p#1#2{\frac{\partial #1}{\partial #2}}
 \p h\alpha = \left(\p \mu\alpha\right)^T\Sigma^{-1}\mu - \mu^T\Sigma^{-1}\p\Sigma\alpha \Sigma^{-1}\mu  + \mu^T \Sigma \p \mu\alpha 
$$
I need to find out the Second derivative wrt $\alpha$. Please help, I searched through net, but couldn't fond any such property.
 A: Assuming that $h = \mu^T \Sigma^{-1}\mu$, as suggested by the derivative, we have: Taking another derivative we apply the product rule as we did for the first and use the chain rule for $\Sigma^{-1}$, in using $\def\pd#1#2{\frac{\partial #1}{\partial #2}}\def\pdd#1#2{\frac{\partial^2 #1}{\partial #2^2}}$ 
$$ \pd{\Sigma^{-1}}{\alpha} = -\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1} $$
So for the first term of $\pd h\alpha$ we have
$$ \pd{}\alpha\left(\pd{\mu^T}\alpha \Sigma^{-1}\mu \right)
   = \pdd{\mu^T}\alpha\Sigma^{-1}\mu - \pd{\mu^T}\alpha \Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu + \pd{\mu^T}\alpha \Sigma^{-1}\pd\mu\alpha 
$$
For the second term
\begin{align*}
&\!\!\!\!\!\!\!\pd{}\alpha\left( \mu^T\Sigma^{-1}\pd\Sigma\alpha \Sigma^{-1}\mu\right)\\
  &= \pd{\mu^T}\alpha \Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu
  - \mu^T\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu
  + \mu^T\Sigma^{-1}\pdd\Sigma\alpha\Sigma^{-1}\mu\\& \ \ {} - \mu^T\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu + \mu^T\Sigma^{-1}\pd\Sigma\alpha \Sigma^{-1}\pd \mu\alpha
\end{align*}
And for the third term
$$ \pd{}\alpha\left(\mu^T \Sigma^{-1}\pd\mu\alpha \right)
   = \pd{\mu^T}\alpha\Sigma^{-1}\pd\mu\alpha - {\mu^T}\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\pd\mu\alpha + {\mu^T} \Sigma^{-1}\pdd\mu\alpha 
$$
Adding everything, we should have 
\begin{align*} \pdd h \alpha &=
  \pdd{\mu^T}\alpha \Sigma^{-1}\mu - \mu^T\Sigma^{-1}\pdd\Sigma\alpha\Sigma^{-1}\mu + \mu^T\Sigma^{-1}\pdd\mu\alpha\\
  &{}+ 2\left(\pd{\mu^T}\alpha \Sigma^{-1}\pd\mu\alpha - \pd{\mu^T}\alpha\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu-\mu^T\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\pd\mu\alpha + \mu^T\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\pd\Sigma\alpha\Sigma^{-1}\mu\right)
\end{align*}
