expression of the 4-tensor $f \otimes g$ in given basis

Question

Let $f$ and $g$ be bilinear functions on $\mathbb{R}^n$ with matrices $a = \{a_{ij}\}$ and $B = \{b_{ij}\}$, respectively. How would I go about finding the expression of the $4$-tensor $f \otimes g$ in the basis$$x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{i_4},\text{ }1 \le i_1, i_2, i_3, i_4 \le n?$$ Any help would be greatly appreciated!

Answer 1

Let $\{e_1, \dots, e_n\}$ be the usual basis for $\mathbb{R}^n$, dual to $x_1, \dots, x_n$. We have $$f\otimes g = \sum_{1 \le i_1, i_2, i_3, i_4 \le n} f \otimes g(e_{i_1}, e_{i_2}, e_{i_3}, e_{i_4})x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{i_4}.$$$($Proof: Let$$H = \sum_{1 \le i_1, i_2, i_3, i_4 \le n} f \otimes g(e_{i_1}, e_{i_2}, e_{i_3}, e_{i_4})x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{i_4}.$$Then $H(e_{j_1}, e_{j_2}, e_{j_3}, e_{j_4}) = f \otimes g(e_{j_1}, e_{j_2}, e_{j_3}, e_{j_4})$. Equality at general $v_1$, $v_2$, $v_3$, $v_4$ then follows by multilinearity of both $H$ and $f \otimes g$.$)$

But$$f \otimes g(e_{i_1}, e_{i_2}, e_{i_3}, e_{i_4}) = f(e_{i_1}, e_{i_2})g(e_{i_3}, e_{i_4}) = a_{i_1 i_2}b_{i_3 i_4}$$so$$f \otimes g = \sum_{1 \le i_1, i_2, i_3, i_4 \le n} a_{i_1 i_2} b_{i_3 i_4} x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{i_4}.$$