Concept

Multilinearity

The property of being linear in each argument separately, with the others held fixed. It is what lets a function of several vectors be expanded over a basis one argument at a time, and it is the first of the three conditions that leave the determinant as the only possibility.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume.

The only function that behaves like a volume

Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.

algebra · determinant

A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both.

The same sum without its minus signs

Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.

algebra · determinant

Named alongside it

The objects these essays reach for when they reach for this one.

Determinant Matrix Permutation Axiom Basis Bounds Cancellation Counting argument Latin square Matching Orientation Parity

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