The excercise I dont understand goes as follow:
We have a symetric positive and definite matrix $A\in R^{dxd}$ and a vector $x\in R^d$. Then we define a function $$f(x)=x^TAx*5$$ Find the gradient of the function $f(x)$
Under tips it is noted tha $f(x)$ is scalar and thus $f(x)$ is equal to its determinant -> $f(x) = det(f(x))$ .
The fact that it is scalar I understand but what does it mean for a function to be equal to its determinant? Where can I read more on this topic.
Apparently there is an easy way to compute the gradient for simetric matrices namely: $$ \frac{\partial det(X^TAX)}{\partial X} = 2*det(X^TAX)AX(X^TAX)^{-1} $$
But I would like to understand what is the theory behind all this. Explanation or references to the theory are most welcome. Thank you.
A scalar valued function to be equal to its determinant means that a scalar can be viewed upon as a $1\times 1$ matrix, and hence it will be equal to its determinant and also trace (which is also used due to the nice property that $\mathrm{tr}(AB) = \mathrm{tr}(BA)$). Also, it is the only eigen value of itself.