If $f : \mathbb{R}^{n} \rightarrow \mathbb{R}$, then I know that $\nabla f(x) \in \mathbb{R}$ and $\nabla^2 f(x) \in \mathbb{R}^{n \times n}$. If I find a point $z$ such that $\nabla f(z) = 0$ and $\nabla^2 f(z) \succeq 0$, then I know that $z$ is a local minimum.
My question concerns generalizing this over matrix spaces. Let $f : \mathbb{R}^{ m \times n} \rightarrow \mathbb{R}$. I know that $\nabla f(x) \in \mathbb{R}^{m \times n}$ and also the Hessian is a rank 4 tensor $\nabla^2 f(x) \in \mathbb{R}^{m \times n \times m \times n}$. What are the equivalent optimality conditions? $\nabla f(z) = 0$ seems clear, but what is the equivalent positive definite-ness condition here? What are some good references for reading about matrix calculus and easy tricks to compute these gradients?