How does a Hessian tensor behave algebraically?
Suppose $H_x$ is the Hessian tensor of a multivariate function $f:\mathbb{R}^n\to\mathbb{R}^m$. This is a third-order tensor and I am told one can think about it as taking two vectors $v, w\in\mathbb{R}^n$ and returning a new vector $H_x(v, w)\in\mathbb{R}^m$.
How does $H_x$ behave algebraically?
Linearity: Given vectors $v_1, v_2, w_1, w_2\in\mathbb{R}^n$ and constants $a, b, c, d\in\mathbb{R}$, how does linearity work? $$ H_x(av_1 + bv_2, cw_1 + dw_2) \overset{?}{=} acH_x(v_1, w_1) + adH_x(v_1, w_2) + bcH_x(v_2, w_1) + bdH_x(v_2, w_2) $$
Matrix Multiplication on $v$: Given $A$ an $n\times n$ matrix can I "take it out"? And in this case, which side do I take it out to? $$ H_x(Av, w) \overset{?}{=} AH_x(v, w) \quad \text{or} \quad \overset{?}{=}H_x(v, w)A \quad \text{or something else?} $$
Matrix Multiplication on $w$: Given $B$ an $n\times n$ matrix multiplying $w$ can I take it out? Where does it go? $$ H_x(v, Bw) \overset{?}{=} BH_x(v, w) \quad \text{or}\quad \overset{?}{=}H_x(v, w)B \quad \text{or something else?} $$
Compatibility with other operations: In many circumstances one has $H_x$ appearing in a product, so it is natural to wonder how it behaves when it is multiplied by tensors of lower order (such as matrices and vectors).
- Vector multiplication: If it is multiplied on the right by a vector, does it get absorbed into one of the arguments? $H_x y \overset{?}{=} H_x(y, \cdot)$ or $\overset{?}{=} H_x(\cdot, y)$?
- Matrix multiplication: If it is multiplied on the right or left by a matrix, does it get absorbed into one of the arguments? $$ A H_x y B \overset{?}{=} H_x(AyB, \cdot) \quad \text{or} \overset{?}{=}A H_x(By, \cdot) $$ How about $$ C(AH_x B) = ? $$
$$H_x(v, w) = \frac{\partial}{\partial t_1}\frac{\partial}{\partial t_2}(f(x + t_1v + t_2w))|_{t_1 = 0}|_{t_2 = 0} = \sum_{j, k = 1}^{n}f_{x_j x_k}v^j w^k.$$ Hence $$(H_x(v, w))_i = H_{f_i}(x)(v, w) = (v, H_{f_i}(x)w).$$ In general, $(Ax, y) = (x, A^Ty)$ for vectors $x, y$ and matrix $A$.