Matrix representation needed

Suppose $$B$$ is an $$N\times N$$ matrix and $$V$$ is an $$N\times 1$$ vector.

I need to find a matrix representation for $$V$$:

$$V_k = \sum_{j}B_{kj}^4 + 2\sum_{i\neq j}\sum_{j}B_{ki}^2B_{kj}^2$$

Previously, I asked a somewhat similar question here: Need to find matrix formulation and I can't find the answer of this one either.

• What do you mean by $B^4_{kj}$? Is it the components of the 4th power of the matrix $(B^4)_{kj}$, or the 4th power of the components of the matrix $(B_{kj})^4$? Are you multiplying matrices or numbers? – mr_e_man Sep 29 '18 at 14:14
• 4th power of the components of matrix B $(B_{kj})^4$, I am multiplying numbers – farhadpti Sep 29 '18 at 19:35

For typing convenience, define $$A=\big(B\odot B\big)$$ where $$\odot$$ represents the elementwise/Hadamard product.
Then the vector can be written in matrix notation as \eqalign{ v &= (A\odot A)\,e + \big(2Ae\big)\odot\big(Ae-{\rm diag}(A)\big) \cr } where $$e$$ is the vector whose elements are all equal to unity.