# Misymmetric matrix

A **skew-symmetrical matrix** (also **antisymmetrical matrix** ) is a matrix that is equal to the negative of its transpose . In a body with characteristics other than two, the skew-symmetrical matrices are exactly the alternating matrices and are therefore often equated with them. Skewly symmetric matrices are used in linear algebra to characterize antisymmetric bilinear forms .

## definition

A matrix is called *skew symmetric* if

applies. In other words: The matrix is skew symmetric if the following applies to its entries:

## example

The matrix is skew symmetrical, there

## properties

### Real skew-symmetric matrices

If is skew symmetric with real entries, then all diagonal entries are necessarily equal to 0. Furthermore, every eigenvalue is purely imaginary or equal to 0.

### Body characteristics not equal to 2

Properties body of characteristic different from 2:

- The entries on the main diagonal are zero.
- The determinant of skew-symmetric matrices with odd dimension
*n*is due to and therefore

- equals zero.
- In general, this does not apply to matrices of even dimension, like the counterexample
- shows. The matrix is obviously skew symmetric, but in general the determinant can be determined as the square of the Pfaff determinant in this case .

- In a body with characteristics other than two, the skew-symmetrical matrices are precisely the alternating matrices . In a body with characteristic two, however, there are skew-symmetrical matrices that are not alternating.

### Vector space

The skew-symmetrical matrices form a vector space of dimension . If the body is, this vector space is called . 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 into the space of the skew-symmetrical matrices is straight with respect to the Frobenius scalar product

The orthogonal complement is the symmetric matrix

### Bilinear forms

The bilinear form to a skew-symmetric matrix is antisymmetric , that is,

for everyone . If the main diagonal entries of a skew-symmetric matrix are all zero (i.e. if the matrix is alternating), then the associated bilinear form is alternating , that is,

for everyone . Conversely, in a finite-dimensional vector space, the representation matrix of an antisymmetric or alternating bilinear form is always skew-symmetric with respect to any base , that is

- ,

where the main diagonal entries are all zero.

### Exponential mapping

The mapping defined by the matrix exponential

is surjective and just describes the exponential mapping on the unit matrix (see also special orthogonal group ).

### Cross product

For the special case , skew-symmetrical matrices can be used to express the cross product as a matrix multiplication . The cross product of two vectors and
can be used as a matrix multiplication of the skew-symmetric *cross product matrix*

can be expressed with the vector :

In this way, a formula with a cross product can be differentiated :

The exponential of the matrix can be represented as follows using Rodrigues' formula

Here is

the orthogonal projection of onto the straight line spanned by , | |

the perpendicular perpendicular to the axis , | |

the vector that arises from rotation by 90 ° around the axis . |

Overall, the formula shows that the exponential of the cross product rotates the vector around the axis defined by , with the norm of as the angular velocity .

## See also

## literature

- DA Suprunenko: Skew-symmetric matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Peter Knabner , Wolf Barth : Lineare Algebra. Basics and applications (= Springer textbook ). Springer Spectrum, Berlin a. a. 2013, ISBN 978-3-642-32185-6 .