Signature
Signature[list]
gives the signature of the permutation needed to place the elements of list in canonical order.
Examples
open all close allBasic Examples (1)
{Signature[{a, b, c}], Signature[{a, c, b}]}Signature[{a, b, b}]Applications (3)
Select[Permutations[{a, b, c, d}], Signature[#] == 1&]Rank-3 totally antisymmetric (Levi-Civita) tensor:
Array[Signature[{##}]&, {3, 3, 3}]Contractions of Levi-Civita tensors:
Table[Sum[Signature[{i, j, k, l}]Signature[{i, j, k, lp}], {i, 4}, {j, 4}, {k, 4}], {l, 4}, {lp, 4}]Properties & Relations (2)
Find components of a 3D cross product:
Table[Sum[Signature[{i, j, k}]{ax, ay, az}[[j]]{bx, by, bz}[[k]], {j, 3}, {k, 3}], {i, 3}]Cross[{ax, ay, az}, {bx, by, bz}]Sum[Signature[{i, j, k}]Array[a, {3, 3}][[1, i]]Array[a, {3, 3}][[2, j]]Array[a, {3, 3}][[3, k]], {i, 3}, {j, 3}, {k, 3}]Compare with built‐in Det:
Det[Array[a, {3, 3}]] == %Possible Issues (2)
The precision of a number influences its ordering:
Signature[{1, 1.}]Signature[{1, 1}]Signature evaluates even for symbolic arguments:
sign = Signature[{a, b, c}]Use Unevaluated to insert an unevaluated Signature:
sign = Unevaluated[Signature[{a, b, c}]];
Table[sign, {a, 3}, {b, 3}, {c, 3}]Use Signature directly inside Table:
Table[Signature[{a, b, c}], {a, 3}, {b, 3}, {c, 3}]Neat Examples (1)
ArrayPlot[1 + Partition[Flatten[Array[Signature[{##}]&, Table[6, {6}]]], 6 ^ 3]]Related Links
MathWorld
The Wolfram Functions SiteText
Wolfram Research (1988), Signature, Wolfram Language function, https://reference.wolfram.com/language/ref/Signature.html.
CMS
Wolfram Language. 1988. "Signature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Signature.html.
APA
Wolfram Language. (1988). Signature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Signature.html