Alternating matrix

In mathematics, an alternating matrix is a square matrix that is skew symmetrical and whose main diagonal entries are all zero . In a field with a characteristic other than two, the second condition follows from the first, which is why alternating matrices are often equated with skew-symmetrical matrices. Alternating matrices are used in linear algebra to characterize alternating bilinear forms . The determinant of an alternating matrix of even size can be given using its Pfaff determinant .

definition

A square matrix with entries from any field is called alternating if ${\displaystyle A\in K^{n\times n}}$ ${\displaystyle K}$

${\displaystyle a_{ij}={-a_{ji}}}$

for and ${\displaystyle i,j=1,\ldots ,n}$

${\displaystyle a_{ii}=0}$

for applies. An alternating matrix is ​​therefore a skew-symmetrical matrix , the main diagonal entries of which are all zero. If the characteristic of the body is not equal to two, then the second condition follows from the first, but this does not apply in a body with characteristic two. ${\displaystyle i=1,\ldots ,n}$

Examples

In the following examples, let the finite field of the remainder classes be modulo , where the remainder class represents the even numbers and the remainder class of the odd numbers . In this body it is true that it has the characteristic . The two alternating matrices of size with entries from this body are ${\displaystyle K={\mathbb {F} }_{2}}$ ${\displaystyle 2}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 1+1=0}$ ${\displaystyle 2}$ ${\displaystyle 2\times 2}$

${\displaystyle {\begin{pmatrix}0&0\\0&0\end{pmatrix}},{\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$

and the total of eight alternating matrices of size are ${\displaystyle 3\times 3}$

${\displaystyle {\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}},{\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}},{\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}},{\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}},{\begin{pmatrix}0&1&1\\1&0&0\\1&0&0\end{pmatrix}},{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}},{\begin{pmatrix}0&0&1\\0&0&1\\1&1&0\end{pmatrix}},{\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}}$.

In this body the skew-symmetrical matrices are precisely the symmetrical matrices that may also have ones on the diagonal.

properties

Bilinear forms

The bilinear form to an alternating matrix is alternating , that is, ${\displaystyle B_{A}(x,y)=x^{T}Ay}$ ${\displaystyle A\in K^{n\times n}}$

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

for everyone . Conversely, in a finite vector space , the representation matrix ${\displaystyle x\in K^{n}}$ ${\displaystyle V}$

${\displaystyle A_{B}=(B(b_{i},b_{j}))}$

an alternating bilinear form with respect to any basis always an alternating matrix. ${\displaystyle B\colon V\times V\to K}$ ${\displaystyle \{b_{1},\ldots ,b_{n}\}}$

rank

The rank of an alternating matrix is always even. There is also a regular matrix , so that after congruence transformation ${\displaystyle r}$ ${\displaystyle A\in K^{n\times n}}$ ${\displaystyle P\in K^{n\times n}}$

${\displaystyle P^{T}AP={\begin{pmatrix}0&I&0\\-I&0&0\\0&0&0\end{pmatrix}}\in K^{n\times n}}$

holds, where is the identity matrix of size . An alternative normal representation is ${\displaystyle I}$ ${\displaystyle {\tfrac {r}{2}}\times {\tfrac {r}{2}}}$

${\displaystyle P^{T}AP={\begin{pmatrix}T&0&0&0\\0&\ddots &0&0\\0&0&T&0\\0&0&0&0\end{pmatrix}}\in K^{n\times n}}$

with exactly blocks of the shape . ${\displaystyle {\tfrac {r}{2}}}$ ${\displaystyle T={\tbinom {~\,0~~1}{-1~~0}}}$

Determinant

Is even, then the can determinant an alternating matrix using the Pfaffian determinant by ${\displaystyle n}$ ${\displaystyle A\in K^{n\times n}}$ ${\displaystyle \operatorname {Pf} (A)}$

${\displaystyle \det A=\operatorname {Pf} (A)^{2}}$

can be specified. If it is odd, then always applies ${\displaystyle n}$

${\displaystyle \det A=0}$.

For further properties of alternating matrices see asymmetric matrix # properties .

See also

• Alternating multilinear form

literature

• Leslie Hogben (Ed.): Handbook of Linear Algebra . CRC Press, 2006, ISBN 978-1-4200-1057-2 .
• Erich Lamprecht : Linear Algebra 2 . Springer, 2013, ISBN 978-3-0348-7680-3 .
• Günter Scheja, Uwe Storch: Textbook of Algebra: Including linear algebra . tape 2 . Vieweg, 1988, ISBN 978-3-322-80092-3 .

Individual evidence

1. ↑ Erich Lamprecht: Lineare Algebra 2 . Springer, 2013, p. 77 .
2. ↑ Günter Scheja, Uwe Storch: Textbook of Algebra: Including linear algebra . 2nd volume. Vieweg, 1988, p. 365 .
3. ↑ ^{a b c} Leslie Hogben (Ed.): Handbook of Linear Algebra . CRC Press, 2006, pp. 12-5 .
4. ↑ Günter Scheja, Uwe Storch: Textbook of Algebra: Including linear algebra . tape 2 . Vieweg, 1988, p. 391 .