Skew symmetry

Skew - symmetry (or antisymmetry in a pair of given arguments) is the property of a mathematical object that is a function of several arguments to change sign (get the factor −1) when rearranging any two arguments.

For example, some square matrices are skew-symmetric (antisymmetric) with respect to the permutation of the indices (i.e., transpose : A ^T = - A , or A _ij = −A _ji ). Obviously, the diagonal elements of such a matrix must be equal to zero.

A rank tensor of at least two may or may not be antisymmetric in some pairs of its indices (channels), or even in all.

Function $f(x_{1},...,x_{n})$ ${\ displaystyle f (x_ {1}, ..., x_ {n})}$ ${\ displaystyle f (x_ {1}, ..., x_ {n})}$ antisymmetric in a pair of arguments $x_{i},\!x_{j},$ ${\ displaystyle x_ {i}, \! x_ {j},}$ ${\ displaystyle x_ {i}, \! x_ {j},}$ if a $f(x_{1},...,x_{i},...,x_{j},...,x_{n})=-f(x_{1},...,x_{j},...,x_{i},...,x_{n}).$ ${\ displaystyle f (x_ {1}, ..., x_ {i}, ..., x_ {j}, ..., x_ {n}) = - f (x_ {1}, ..., x_ {j}, ..., x_ {i}, ..., x_ {n}).}$ ${\ displaystyle f (x_ {1}, ..., x_ {i}, ..., x_ {j}, ..., x_ {n}) = - f (x_ {1}, ..., x_ {j}, ..., x_ {i}, ..., x_ {n}).}$ For example, the antisymmetric function $f(x,y)=x-y.$ ${\ displaystyle f (x, y) = xy.}$ ${\ displaystyle f (x, y) = x-y.}$

A binary operation is skew-symmetric if its result changes sign when the operands are rearranged. Examples are a subtraction operation, a vector product operation, Poisson brackets , a commutator . A ternary operation can also be skew-symmetric (for example, a mixed product of vectors is skew-symmetric with respect to any pair of operands).

An absolutely skew-symmetric object changes sign when rearranging any two arguments (indices). Some objects may be skew-symmetric in one pair of indices and not possess skew-symmetry in other pairs.

Skew symmetry

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