Skew-symmetric matrix

A skew-symmetric matrix ( also anti-symmetric matrix ) is a matrix that is equal to the negative of its transpose.

Definition

A matrix is called skew-symmetric matrix, if

Applies. Equivalent can be the skew-symmetry also define component-wise: The matrix is ​​skew-symmetric if and only if for all entries:

Example

The matrix is ​​skew-symmetric, since

Properties

Real skew-symmetric matrices

Is skew-symmetric with real entries, so all diagonal entries are necessarily equal to 0 Furthermore, all eigenvalues ​​are purely imaginary or equal to 0

Body characteristic different from 2

Properties for field of characteristic different from 2:

Vector space

The skew-symmetric matrices form a vector space of dimension. Is the body so one calls with this vector space. The name comes from the fact that this vector space is the Lie algebra of the Lie group ( Special orthogonal group ).

The orthogonal projection from the space of the matrices in the space of skew-symmetric matrices with respect to the straight Standardskalarprodukts

, The orthogonal group is the symmetric matrix

Exponential

The figure

Converges, is surjective and just describes the exponential map to the unit matrix ( see also Special orthogonal group ).

Cross product

For the special case of skew-symmetric matrices can be used to define a multiplication of vectors. This is because only in this case the skew-symmetric matrix just has degrees of freedom. In this vector multiplication is the cross product.

The cross product of and may be expressed as matrix multiplication by using

Here is the skew-symmetric matrix

Using this representation, it is easier to differentiate the cross product:

The exponential of the matrix, by means of Rodrigues formula shown below are

• Denotes the projection of on the plane spanned by axis.
• Is perpendicular to
• Describes the 90 ° rotated vector

With these notations, the above formula can be simplified as follows:

Overall, the formula that is rotated by the axis defined by the exponential of the cross product of the vector with the standard of the angular velocity.