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I have a function $f \colon \mathbb{R}^{d \times d} \times \mathbb{R}^d \to \mathbb{R}$ defined by $$ f(A, c) = (z - c)^T A (z - c) - 1 $$ where $A \in \mathbb{R}^{d \times d}$ is symmetric positive definite, $c \in \mathbb{R}^d$, and $z_i \in \mathbb{R}^d$ is a given constant. This function is used in defining an optimization problem, and so it is of interest to find wether or not this function is in $C^1$ and $C^2$. I know that for simpler functions, i.e. 1-dimensional functions $g \colon \mathbb{R} \to \mathbb{R}$, this simply involves showing that the function is once and twice differentiable. However, this problem has me confused. How would I even go about computing the gradient of a function like this?

All help is appreciated. If anything is unclear or further clarification is needed, please let me know.

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    $\begingroup$ If we're restricting $A$ to be symmetric and positive definite, then the domain of $f$ is not $\Bbb R^{d\times d}\times \Bbb R^d$. $\endgroup$ – Arthur Mar 11 at 17:04
  • $\begingroup$ You are of course correct. Is there a standard notation for the set of symmetric and positive definite matrices? I also don't think the restriction on A is of relevance for the question, so I could remove it if this makes the question more clear. $\endgroup$ – Erik André Mar 11 at 17:30
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    $\begingroup$ Your function is a polynomial (in $d^2+d$ variables). All partial derivatives of $f$ (of any order) are also polynomials and in particular continuous. The function $f$ is thus of class $C^\infty$. $\endgroup$ – Amitai Yuval Mar 11 at 18:43

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