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Dual-Manipulator Testing Technique
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
According to the theories of Lie group and Lie algebra, the exponential matrix of a skew symmetric matrix is an orthogonal matrix. In the inverse function, the logarithm matrix of an orthogonal matrix is a skew symmetric matrix. Skew symmetric matrix is a square matrix whose transpose plus itself is equal to null matrix. In the real number domain, a skew symmetric matrix represents an identity matrix mapped from the tangent space to the real orthogonal group O(n). This is the so-called special orthogonal Lie algebra. Therefore, skew symmetric matrix can be regarded as an infinitesimal rotation, and its space constitutes the Lie algebra O(n) of the Lie group O(n). It is easy to prove that the addition and subtraction of two skew symmetric matrices will still produce a skew symmetric matrix.
Matrices and linear transformations
Published in Alan Jeffrey, Mathematics, 2004
A skew-symmetric matrix is one in which the elements obey the rule aij = –aji, so that the leading diagonal must contain zeros, while the pattern of numbers has a reflection symmetry about the principal diagonal but with a reversal of sign. A typical skew-symmetric matrix of order 3 × 3 is: [015−10−3−530].
Minimal realisations of odd transfer functions for first-degree nD systems
Published in International Journal of Control, 2020
(Cameron, 1991, P105, Theorem 8.1):The determinant of a skew-symmetric matrix of odd size is zero.There is a unique polynomial Pf(A) in the indeterminates ai, j for 1 ≤ i < j ≤ 2n, having the propertiesif A is a 2n × 2n skew-symmetric matrix with (i, j)-entry ai, j for 1 ≤ i < j ≤ 2n, then Pf(A) contains the term a1, 2a3, 4⋅⋅⋅a2n − 1, 2n with coefficient +1.