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Using e.g. the Gauss elimination, we can diagonalize the matrix $(v_{kl})$, that is, write $$v_{kl}=\sum_{r=1}^d a_r s_{rk}t_{rl}$$ for some real $a_r,s_{rk},t_{rl}$ and all $k,l$; the computational complexity (CC) of this diagonalization is $O(d^3)$; cf. e.g. this source. Now we can write $$u_{ij}=\sum_r a_r\,Ex_ix_jX_rY_r,$$ where X_r:=\sum_k s_{rk}x_k,\...