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A skew-symmetric matrix is a square matrix whose transpose equals its negative. Learn about its properties, vector space structure, determinant, cross product, spectral theory and applications to infinitesimal rotations.
28 paź 2024 · An antisymmetric matrix is a square matrix that satisfies A=-A^ (T), where A^ (T) is the transpose. Learn how to test, decompose and apply antisymmetric matrices in linear algebra and Wolfram Language.
Learn about the properties and theorems of antisymmetric matrices, and how to define and calculate the pfaffian of an even-dimensional complex antisymmetric matrix. The notes also cover the real normal form, the congruence classes, and the determinant of antisymmetric matrices.
Learn what an antisymmetric matrix is, how to recognize it, and how to calculate its determinant and eigenvalues. Also, find out how to decompose any square matrix into a symmetric and an antisymmetric matrix.
Here are the definitions: (1) A matrix A A is symmetric that satisfies the identity A =AT A = A T; (2) a matrix B B is anti-symmetric that satisfies the identity B = −BT B = − B T, and (3) a matrix C C is diagonal if it's a square matrix with the entries outside of the diagonal all zero.
Learn what an antisymmetric matrix is, how to calculate it from a square matrix, and how it relates to its transpose and symmetric part. See examples of antisymmetric matrices and their properties, such as the zero diagonal and opposite elements.
Properties of antisymmetric matrices. Let M be a complex d × d antisymmetric matrix, i.e. MT = −M. Since. det M = det (−MT) = det (−M) = (−1)d det M , (1) it follows that det M = 0 if d is odd. Thus, the rank of M must be even. In these notes, the rank of M will be denoted by 2n. If d det M = 0.
