What is the term to describe matrices created using the derivatives of a vector of functions?

Question

For example, given a vector of functions, $F = [f_1, f_2, \dots, f_n]$, and variables, $X = [x_1, x_2, \dots, x_n]$

The matrix created using the $1$st derivative of $F$ wrt $X$ is called the Jacobian Matrix. The matrix created using the $2$nd derivative of $F$ wrt $X$ is called the Hessian Matrix.

Is there a name for the matrix created using the $n$th derivative? basically a more general name to use instead of $J$ for $n = 1$ and $H$ for $n = 2$.

Answer 1

The point is that a matrix is a useful tool to represent a collection of $nm$ objects with two indexes $i,j$. For example, if you have a differentiable $$F:\mathbb{R}^n\to\mathbb{R}^m$$ then partial derivatives are $\partial F_i/\partial x_j$, indexed by $i,j$. Therefore you can represent them with the usual Jacobian matrix. But notice that, for second derivatives, you have in fact three indexes: $$\frac{\partial^2F_i}{\partial x_j x_k}$$ Therefore you cannot represent them with a usual matrix (e.g., at row $i$ and column $j$ should one put $\frac{\partial^2F_i}{\partial x_j x_\color{red}{1}}$ or $\frac{\partial^2F_i}{\partial x_j x_\color{red}{2}}$? In some sense you need a sort of "three-dimensional matrix"). The case of Hessian matrix you referred to is possible for a funcion with just one real value: $$f:\mathbb{R}^n\to\mathbb{R}$$ In that case, second derivatives are just $$\frac{\partial^2 f}{\partial x_j x_k}$$ depending only from two indexes. For this case, first derivatives have only one index: $\partial f/\partial x_j$, therefore you can represent them just with a vector (usually called gradient).

Thus, in general, for $m^{th}$-order partial derivatives you have a multi-indexed collection. In particular, Hessian and Jacobian are the only cases in which you can use a matrix to represent partial derivatives, since they are the only cases in which there are exactly two indexes, i.e.

• Hessian: second derivatives of a real-valued function
• Jacobian: first derivatives of a vector-valued function

You could notice that in fact they are quite the same thing, since the Hessian matrix of the real-valued function $f$ is precisely the Jacobian matrix of the vector-valued function $F=\nabla f=(\partial f/\partial x_1,\dots ,\partial f/\partial x_n)$