![]() It is also necessarily linear in each variable separately, which can also be seen geometrically. Such a function is necessarily alternating. Not an initial condition, but related: c(n, k) 0 for k > n since the permutation of n with the most cycles is (1)(2) (n). v_n calculates the signed volume of the parallelpiped given by the vectors v_1. since every permutation of n must have at least one cycle. The determinant of a matrix with columns v_1. From a geometric persepective, that is how alternating functions come into play. If you swap two vectors that reverse the orientation of the parellelpiped, so you should get the negative of the previous answer. In R^n it is useful to have a similar function that is the signed volume of the parallelpiped spanned by n vectors. If you swap x and y you get the negative of your previous answer. It cares about the direction of the line from x to y and gives you positive or negative based on that direction. It really gives you a bit more than length because is a signed notion of length. On the real line function of two variables (x,y) given by x-y gives you a notion of length. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. There is a geometric side, which gives some motivation for his answer, because it isn't clear offhand why multilinear alternating functions should be important. The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. I think Paul's answer gets the algebraic nub of the issue. There is a close connection between the space of alternating $k$-linear functions and the $k$-order wedge product of a space, so I could have very similarly developed the determinant based on the wedge product, but alternating $k$-linear functions are easier conceptually. Lempel Permutation graphs and transitive graphs J. ![]() In particular that $\det(MN) = \det(M)\det(N)$. A graph is a permutation graph iff it has an intersection model consisting of straight lines (one per vertex) between two parallels. Certain properties of determinants that are difficult to prove from the Liebnitz formula are almost trivial from this definition. ![]() This is only one of many possible definitions of the determinant.Ī more "immediately meaningful" definition could be, for example, to define the determinant as the unique function on $\mathbb R^f \in A^n(V)$$Īll the properties of determinants, including the permutation formula can be developed from this.
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