An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity. (1) where is the matrix transpose. For example, (2)

skew-symmetric
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition.

A Skew Symmetric Matrix or Anti-Symmetric Matrix is a square matrix whose transpose is negative to that of the original matrix. If the entry in the ith row and jth column of a matrix is a[i][j], i.e. if A = (a[i][j]) then the skew symmetric condition is -A = -a[j][i].

Fact: A matrix has orthogonal eigenvectors exactly when AAT = AT A; i.e. when A commutes with its transpose. This is true of symmetric, skew symmetric and orthogonal matrices. The matrix is diagonalizable if it has 3 independent eigenvectors.

: relating to or being a relation (such as “is a subset of”) that implies equality of any two quantities for which it holds in both directions the relation R is antisymmetric if aRb and bRa implies a = b.

Antisymmetric function – Antisymmetrische Funktion Important special cases antisymmetric functions are antikommutative links and alternating multilinear forms . In quantum mechanics , fermions are precisely those particles whose wave function is antisymmetric with respect to the exchange of particle positions.

A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. For a general tensor U with components and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: (symmetric part)

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length.

To prove an antisymmetric relation, we assume that (a, b) and (b, a) are in the relation, and then show that a = b. To prove that our relation, R, is antisymmetric, we assume that a is divisible by b and that b is divisible by a, and we show that a = b.