Skew symmetric matrices

note about skew symmetric matrices

Skew symmetric matrices

Definition

\[ [a]_{\times} = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0\\ \end{bmatrix} \]

The matrix [x] is a skew-symmetric matrix that \[ [x] = -[x]^{T} \] The set of all \(3 \times 3 \) skew-symmetric matrices is called \(so(3)\) *The set of skew-symmetric matrices so(3) is called Lie Algebra of the Lie Group. It consists of all possible \(\dot{R}\) when \(R = I\).

Properties

Given any \(\omega \in \mathbb{R}^3\) and \(R \in SO(3)\), the following always holds \[ R[\omega]R^T = [R\omega] \]

1. The sum of two skew-symmetric matrices is skew-symmetric
2. A scalar multiple of a skew-symmetric is skew-symmetric
3. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
4. If A is a real skew-symmetric matrix and \(\lambda\) is a real eigenvalue, then \(\lambda = 0\), i.e., the nonzero eigenvalues of a skew-symmetric matrix are non-real.
5. If A is a real skew-symmetric matrix, then I+A is invertible, where I is the identity matrix
6. If A is a skew-symmetric matrix then \(A^2\) is a symmetric negative semi-definite matrix

Vector space

As a result of the first properties above, the set of all skew-symmetric matrices of a fixed size forms a Vector space

square properties

\([\hat{\omega}]^{3}=-[\hat{\omega}]\) \[ [\hat{\omega}]^{4} = -[\hat{\omega}]^{2} \]