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If one calculates the product of diagonal elements of the $U$ matrix in a $LUP$ factorization of a given matrix $A$, one can calculate the determinant of $A$. Also it is known that $LUP$ factorization can be reduced to a series of matrix multiplications which are bilinear forms. Does that mean determinant calculation (multilinear form) can be reduced to a bilinear calculation?

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  • $\begingroup$ Sorry for my ignorance, but what is bilinear calculation? (I know what a bilinear form is.) $\endgroup$ Nov 5, 2010 at 2:16
  • $\begingroup$ Hi Tusyoshi I was implying the same. $\endgroup$
    – Turbo
    Nov 5, 2010 at 2:35
  • $\begingroup$ Obviously the deterinant is not a bilinear form in the elements of the matrix (no matter how you divide the elements into two sets), if that is what you are asking. $\endgroup$ Nov 5, 2010 at 3:06
  • $\begingroup$ Hi Tsuyoshi. Yes I know that determinant is a multilinear form. However using only a sequence of matrix multiplications one can reduce to $LUP$. From which taking the product of diagonals of $U$ suffice to find the determinant. Multiplication is a bilinear map. I was curious then may be the determinant is actually a product of bilinear forms. $\endgroup$
    – Turbo
    Nov 5, 2010 at 3:24
  • $\begingroup$ I feel this is just a tautological question. $\endgroup$
    – Turbo
    Nov 5, 2010 at 3:25

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Yes, because (equivalently) determinants can be computed via row reduction. Those steps exploit only the bilinearity and the antisymmetry of the determinant.

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