antisümmeetrilised

Antisümmeetrilised refers to a property of certain mathematical objects, most commonly matrices and tensors, indicating a specific type of symmetry reversal. In the context of matrices, an antisümmeetriline matrix, also known as a skew-symmetric matrix, is a square matrix A for which its transpose is equal to its negative. This can be expressed mathematically as A^T = -A.

A key characteristic of antisümmeetrilised matrices is that all elements on their main diagonal must be zero. This is because for any diagonal element a_ii, the condition a_ii = -a_ii must hold, which is only possible if a_ii = 0. Furthermore, for any pair of off-diagonal elements a_ij and a_ji, the relationship a_ij = -a_ji is satisfied. This means that elements symmetrically positioned across the main diagonal have opposite signs.

Antisümmeetrilised matrices appear in various areas of mathematics and physics. For example, they are fundamental in the study of Lie algebras and play a role in representing rotations and angular momentum in quantum mechanics. In linear algebra, properties such as determinants of antisümmeetrilised matrices have specific behaviors; for instance, the determinant of an antisümmeetriline matrix of odd dimension is always zero. The concept extends to higher-order tensors, where antisümmeetrilisus describes how the tensor's components change sign under the swapping of specific indices.