I have the following:
$$b^T a^{-1} b$$ what is the gradient wrt to $a$. $a$ is matrix and $b$ is vector. Basically I should take the derivative with respect to $a$. Is it correct that it equals to:
$$-ba^{-2}b^T$$ or there is a better way to represent? If someone give me a step by step instruction I would appreciate.
Let $f:a\in M_n(\mathbb{R})\rightarrow b^Ta^{-1}b$. The derivative is: $Df_a:H\in M_n(\mathbb{R})\rightarrow -b^Ta^{-1}Ha^{-1}b=trace(-a^{-1}bb^Ta^{-1}H)$ and the gradient is $\nabla f(a)=-(a^{-1})^Tbb^T(a^{-1})^T$.