Skew-Symmetric Matrix

A matrix that equals the negative of its own transpose; in 3×3 form it turns a cross product into matrix multiplication.

A skew-symmetric matrix satisfies Aᵀ = −A, and its diagonal entries are all zero. The most common case in robotics is the 3×3 version: given a vector ω = (ω₁, ω₂, ω₃), construct the matrix [ω] = [[0, −ω₃, ω₂], [ω₃, 0, −ω₁], [−ω₂, ω₁, 0]], and then [ω]v is exactly equal to the cross product ω×v. Turning a vector into this matrix is called the hat operation (written ω^ or [ω]×), and recovering the vector from the matrix is called vee. Its importance is that all 3×3 skew-symmetric matrices together form so(3), the Lie algebra of the rotation group SO(3), which can be thought of as the space of infinitesimal rotations; taking the matrix exponential of one gives a rotation matrix, and the closed-form result is Rodrigues' rotation formula. It comes up repeatedly when deriving angular velocity, the derivative of a rotation matrix, and Jacobian matrices.

ExampleTake ω = (0,0,1) (a unit angular velocity about the z-axis) and v = (1,0,0): [ω]v = (0,1,0), matching the cross product ω×v — meaning the point on the x-axis is, at this instant, moving in the y direction.

Also called

Cross-Product Matrix, Hat Operator, [ω]×

Related

Rodrigues' Rotation Formula · Rotation Matrix · Special Orthogonal Group SO(3) · Lie Group · Exponential Map · Screw Theory

Sources

Wikipedia: Skew-symmetric matrix
Wikipedia: Rodrigues' rotation formula