What is the second derivative of $Tr(A^T(\alpha)BA(\alpha))$?

Question

What is the second derivative $\frac{d^2}{d\alpha}Tr(A^T(\alpha)BA(\alpha))$? Here, $B$ is square matrix and $A(\alpha)$ is a parameter dependent matrix that is rectangular. All entries of $B, A$ are always real.

Answer 1

You asked about the first derivative of this function in your previous question

What is the derivative of $\frac{d}{d\alpha}Tr(A^T(\alpha)BA(\alpha))$? $$ $$ Starting with that result $$\eqalign{ \frac{df}{d\alpha} &= \Big((B+B^T)A:\frac{dA}{d\alpha}\Big) \cr\cr }$$ Finding the second derivative is very straight-forward $$\eqalign{ \frac{d^2f}{d\alpha^2} &= \Big((B+B^T)\frac{dA}{d\alpha}:\frac{dA}{d\alpha} + (B+B^T)A:\frac{d^2A}{d\alpha^2}\Big) \cr &= {\rm tr}\Big((B+B^T)\frac{dA}{d\alpha}\frac{dA}{d\alpha}^T + (B+B^T)A\,\frac{d^2A^T}{d\alpha^2}\Big)\cr }$$