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 ${\displaystyle A}$

${\displaystyle A^{T}=-A}$

applies. In other words: The matrix is skew symmetric if the following applies to its entries: ${\displaystyle A}$

${\displaystyle a_{ij}=-a_{ji}\qquad \forall i,j\in \{1,\ldots ,n\}}$

example

The matrix is skew symmetrical, there ${\displaystyle A={\begin{pmatrix}0&7&23\\-7&0&-4\\-23&4&0\end{pmatrix}}}$ ${\displaystyle A^{T}={\begin{pmatrix}0&-7&-23\\7&0&4\\23&-4&0\end{pmatrix}}=-A}$

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. ${\displaystyle A\in \mathbb {R} ^{n\times n}}$

Body characteristics not equal to 2

Properties body of characteristic different from 2: ${\displaystyle K}$

• The entries on the main diagonal are zero.
• The determinant of skew-symmetric matrices with odd dimension n is due to and therefore ${\displaystyle A^{T}=-A\,}$

${\displaystyle \det(A)=\det(A^{T})=\det(-A)=(-1)^{n}\,\det(A)=-\det(A)}$

equals zero.

In general, this does not apply to matrices of even dimension, like the counterexample

${\displaystyle A={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$

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 . ${\displaystyle \det(A)=1.}$

• 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 ). ${\displaystyle {\tfrac {n(n-1)}{2}}}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle {\mathfrak {so}}(n)}$ ${\displaystyle \operatorname {SO} (n)}$

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

${\displaystyle {\begin{matrix}\operatorname {Pr} :&\mathbb {R} ^{n\times n}&\to &{\mathfrak {s}}{\mathfrak {o}}(n)\\&A&\mapsto &{\frac {1}{2}}(A-A^{T})\end{matrix}}}$

The orthogonal complement is the symmetric matrix

${\displaystyle A-\operatorname {Pr} (A)={\frac {1}{2}}(A+A^{T}).}$

Bilinear forms

The bilinear form to a skew-symmetric matrix is antisymmetric , that is, ${\displaystyle B_{A}(x,y)=x^{T}Ay}$ ${\displaystyle A\in K^{n\times n}}$

${\displaystyle B_{A}(x,y)=-B_{A}(y,x)}$

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, ${\displaystyle x,y\in K^{n}}$ ${\displaystyle A}$ ${\displaystyle B_{A}}$

${\displaystyle B_{A}(x,x)=0}$

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 ${\displaystyle x\in K^{n}}$ ${\displaystyle V}$ ${\displaystyle A_{B}=(B(b_{i},b_{j}))}$ ${\displaystyle B\colon V\times V\to K}$ ${\displaystyle \{b_{1},\ldots ,b_{n}\}}$

${\displaystyle (A_{B})^{T}=-A_{B}}$,

where the main diagonal entries are all zero. ${\displaystyle A_{B}}$

Exponential mapping

The mapping defined by the matrix exponential

${\displaystyle {\begin{matrix}\exp :&{\mathfrak {s}}{\mathfrak {o}}(n)&\to &\operatorname {SO} (n)\\&A&\mapsto &\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}A^{n}\end{matrix}}}$

is surjective and just describes the exponential mapping on the unit matrix (see also special orthogonal group ). ${\displaystyle I_{n}}$

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 ${\displaystyle n=3}$ ${\displaystyle a\in \mathbb {R} ^{3}}$ ${\displaystyle b\in \mathbb {R} ^{3}}$

${\displaystyle [a]_{\times }={\begin{pmatrix}0&-a_{3}&a_{2}\\a_{3}&0&-a_{1}\\-a_{2}&a_{1}&0\end{pmatrix}}}$

can be expressed with the vector : ${\displaystyle b}$

${\displaystyle a\times b=[a]_{\times }\cdot b.}$

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

${\displaystyle {\frac {\partial }{\partial b}}(a\times b)=[a]_{\times }}$

The exponential of the matrix can be represented as follows using Rodrigues' formula ${\displaystyle [a]_{\times }}$

${\displaystyle {\begin{aligned}\exp(t[a]_{\times })v&={\tfrac {\langle a,v\rangle }{\|a\|^{2}}}a+\left(v-{\frac {\langle a,v\rangle }{\|a\|^{2}}}a\right)\cos(\|a\|\,t)+\left({\frac {1}{\|a\|}}a\times v\right)\sin(\|a\|\,t)\\&=v_{a}+v_{0}\cdot \cos(\|a\|\,t)+v_{1}\cdot \sin(\|a\|\,t).\end{aligned}}}$

Here is

${\displaystyle v_{a}:={\frac {\langle a,v\rangle }{\|a\|^{2}}}a}$	the orthogonal projection of onto the straight line spanned by , ${\displaystyle v}$ ${\displaystyle a}$ ${\displaystyle L_{a}}$
${\displaystyle v_{0}:=v-v_{a}}$	the perpendicular perpendicular to the axis , ${\displaystyle v}$ ${\displaystyle L_{a}}$
${\displaystyle v_{1}:={\frac {1}{\|a\|}}a\times v_{0}}$	the vector that arises from rotation by 90 ° around the axis . ${\displaystyle v_{0}}$ ${\displaystyle L_{a}}$

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 . ${\displaystyle v}$ ${\displaystyle a}$ ${\displaystyle a}$

See also

• Symmetrical matrix
• Schiefhermitean matrix

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 .