Convexity of $t\mapsto b'e^{-A-Bt}b$

Question

Let $b$ be some given vector and $A,B$ arbitrary symmetric matrices, and $A$ is positive definite. Is the function $$t\mapsto b'e^{-A-Bt}b$$ convex?

This is related to another question: Convexity of $x\mapsto \mathrm{tr}(e^{-E\langle a,x\rangle}bb')$

Answer 1

If $AB=BA$, then the function is convex; otherwise, it's false. See, for example

$b=[45,-14]^T,A=\begin{pmatrix}13.84&5.64\\5.64&2.44\end{pmatrix},B=\begin{pmatrix}-0.26&-0.63\\-0.63&-0.23\end{pmatrix}$ on $t\in [5,15]$.