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principal minors

Striking out rows and columns intersecting on a diagonal will always leave some surviving elementary determinants, until so many have been stricken out that it is no longer possible to form a complete determinant. The coefficient of the elementary determinants will depend on how much of the remaining elementary determinant can be salvaged.

In general, the determinant of the de Bruijn factorizes, so that

where the factors are determinants obtained by varying the indices i and j with fixed To get the minor of the hz,hz diagonal element, we need to consider two cases. If this pivot does not belong to any subdeterminant, it can be ignored. If it does, we drop the z,z term from the product, replacing it by the internal minor from within the omitted factor. Thus

Harold V. McIntosh