Monomial matrix

In mathematics, a monomial matrix or generalized permutation matrix is a square matrix in which exactly one entry is not equal to zero in each row and each column. Monomial matrices thus represent a generalization of ordinary permutation matrices in which exactly one entry per row and column is equal to one. The regular monomial matrices form the monomial group with the matrix multiplication as a link . Monomial matrices are used in geometry , group theory and coding theory , among others .

definition

A monomial matrix is ​​a square matrix in which exactly one entry per row and column is not equal . This is generally the zero element of an underlying ring . Every monomial matrix can be described as a product ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle R}$ ${\displaystyle G\in R^{n\times n}}$

${\displaystyle G=P\cdot D}$   or.   ${\displaystyle G=D\cdot P}$

from a permutation matrix and a diagonal matrix. If commutative , then the two representations are equivalent: in the first representation the diagonal entries of each correspond to the column entries unequal to , in the second representation each correspond to the row entries unequal ; the two permutation matrices agree. ${\displaystyle P\in R^{n\times n}}$ ${\displaystyle D\in R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle D}$ ${\displaystyle 0}$ ${\displaystyle G}$ ${\displaystyle 0}$

example

An example of an integer monomial matrix is

${\displaystyle G={\begin{pmatrix}0&6&0\\0&0&4\\2&0&0\end{pmatrix}}}$,

because it applies

${\displaystyle G={\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}\cdot {\begin{pmatrix}2&0&0\\0&6&0\\0&0&4\end{pmatrix}}={\begin{pmatrix}6&0&0\\0&4&0\\0&0&2\end{pmatrix}}\cdot {\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}}$.

Special cases

Special classes of monomial matrices are:

• Permutation : here the matrix is the identity matrix ${\displaystyle D}$
• Signed permutation matrices : here the diagonal entries of the matrix are all either or ${\displaystyle D}$ ${\displaystyle +1}$ ${\displaystyle -1}$
• Generalized permutation matrices: here the diagonal entries of the matrix are all -th roots of unity for a ${\displaystyle D}$ ${\displaystyle m}$ ${\displaystyle m\in \mathbb {N} }$
• Diagonal matrices : here the matrix is the permutation matrix of the identical permutation ${\displaystyle P}$
• Antidiagonal matrices : here the matrix is the permutation matrix of the reverse identical permutation ${\displaystyle P}$

properties

number

If the support set of the ring is finite with elements, then the number of different monomial matrices is the size ${\displaystyle R}$ ${\displaystyle k}$ ${\displaystyle n\times n}$

${\displaystyle k^{n}\cdot n!=k\cdot 2k\cdot \ldots \cdot nk}$,

for there are various permutation matrices of size and possible choices for non-zero entries. ${\displaystyle n!}$ ${\displaystyle n\times n}$ ${\displaystyle k^{n}}$ ${\displaystyle n}$

product

The product of two monomial matrices is again a monomial matrix because it holds ${\displaystyle G,G'\in R^{n\times n}}$

${\displaystyle G\cdot G'=(P\cdot D)\cdot (P'\cdot D')=(P\cdot P')\cdot ({\bar {D}}\cdot D')}$,

where is the diagonal matrix that results from swapping rows and columns according to the underlying permutation . The set of monomial matrices of fixed size therefore forms a semi-group with the matrix multiplication as a link . ${\displaystyle {\bar {D}}=(P')^{T}\cdot D\cdot P'}$ ${\displaystyle D}$ ${\displaystyle P'}$

Inverse

A monomial matrix can be inverted if its entries are not equal to zero units (multiplicatively invertible elements) in the ring . If a body or a sloping body , then all elements not equal to zero are units and thus all monomial matrices are invertible. The inverse matrix of results in ${\displaystyle G\in R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle G}$

${\displaystyle G^{-1}=(P\cdot D)^{-1}=D^{-1}\cdot P^{-1}=D^{-1}\cdot P^{T}}$,

where the permutation matrix is ​​the inverse permutation and the diagonal matrix consisting of the multiplicative inverses of the diagonal entries of . The regular monomial matrices with the matrix multiplication as a link form the monomial group , a subgroup of the general linear group . ${\displaystyle P^{-1}}$ ${\displaystyle D^{-1}}$ ${\displaystyle D}$ ${\displaystyle \operatorname {M} (n,R)}$ ${\displaystyle \operatorname {GL} (n,R)}$

Determinant

The determinant of a monomial matrix with entries from a commutative ring arising after the determinants product set to ${\displaystyle R}$

${\displaystyle \det(G)=\det(P)\cdot \det(D)=\operatorname {sgn} (\pi )\cdot d_{11}\cdot \ldots \cdot d_{nn}}$,

where is the sign of the associated permutation and are the diagonal elements of . ${\displaystyle \operatorname {sgn} (\pi )}$ ${\displaystyle P}$ ${\displaystyle \pi }$ ${\displaystyle d_{11},\ldots ,d_{nn}}$ ${\displaystyle D}$

Real monomial matrices

The inverse of a real monomial matrix is created by transposing and forming the reciprocal values ​​of all entries not equal to zero, for example ${\displaystyle G\in \mathbb {R} ^{n\times n}}$

${\displaystyle {\begin{pmatrix}0&6&0\\0&0&4\\2&0&0\end{pmatrix}}^{-1}={\begin{pmatrix}0&0&{\tfrac {1}{2}}\\{\tfrac {1}{6}}&0&0\\0&{\tfrac {1}{4}}&0\end{pmatrix}}}$.

The inverse of a nonnegative monomial matrix is ​​therefore always nonnegative again. The inverse is even true and a regular nonnegative matrix whose inverse is also nonnegative is monomial. Since the product of two non-negative monomial matrices is also non-negative again, the non-negative monomial matrices form a subgroup of the monomial group.

use

In geometry , monomial matrices whose non-zero entries only consist of the numbers or have a special meaning. The group of these signed permutation matrices is isomorphic to the hyperoctahedron group , the symmetry group of a hypercube or cross polytope in -dimensional space. ${\displaystyle +1}$ ${\displaystyle -1}$ ${\displaystyle n}$

In group theory , monomial matrices play an important role in the monomial representation of finite groups .

In coding theory , two linear codes are called equivalent to each other if there is a monomial matrix that converts both codes into one another.

literature

• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 978-0-521-83940-2 .
• Christian Voigt, Jürgen Adamy: Collection of formulas for matrix calculation . Oldenbourg, 2007, ISBN 3-486-58350-6 .

Web links

• DA Suprunenko: Monomial matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• GrafZahl: Monomial matrix . In: PlanetMath . (English)

Individual evidence

1. ^ Christian Voigt, Jürgen Adamy: Collection of formulas for the matrix calculation . Oldenbourg Verlag, 2007, ISBN 3-486-58350-6 , p. 85 .
2. ^ Roger A. Horn, Charles Johnson: Matrix analysis . Cambridge University Press, 2013, ISBN 978-0-521-83940-2 , pp. 33 .
3. ↑ January Okniński: Semi Groups of matrices . World Scientific, 1998, ISBN 978-981-02-3445-4 , pp. 76 .
4. ↑ Tadeusz Kaczorek: Positive 1D and 2D system . Springer, 2012, ISBN 978-1-4471-0221-2 , pp. 1-2 .
5. ^ Michael Field: Lectures on Bifurcations, Dynamics and Symmetry . CRC Press, 1996, ISBN 978-0-582-30346-1 , pp. 12-13 .
6. ^ VL Popov: Monomial representation . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
7. ↑ Wolfgang Willems: Coding Theory and Cryptography . Springer, 2011, ISBN 978-3-7643-8612-2 , pp. 25 .