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In [[mathematics]], a '''generalized permutation matrix''' (or '''monomial matrix''') is a [[matrix (mathematics)|matrix]] with the same nonzero pattern as a [[permutation matrix]], i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is
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:<math>\begin{bmatrix}
0 &  0 & 3 & 0\\
0 & -2 & 0 & 0\\
1 &  0 & 0 & 0\\
0 &  0 & 0 & 1\end{bmatrix}.</math>
 
== Structure ==
An [[invertible matrix]] ''A'' is a generalized permutation matrix if and only if it can be written as a product of an [[invertible matrix|invertible]] [[diagonal matrix]] ''D'' and an (implicitly [[invertible matrix|invertible]]) [[permutation matrix]] ''P'': i.e.,
 
:<math> A=DP. </math>
 
===Group structure===
The set of ''n''&times;''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'',''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a [[normal subgroup]]. Indeed, the generalized permutation matrices are the [[normalizer]] of the diagonal matrices, meaning that the generalized permutation matrices are the ''largest'' subgroup of GL in which diagonal matrices are normal.
 
The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''<sup>&times;</sup> and ''S''<sub>''n''</sub>. Concretely, this means that it is the [[semidirect product]] of Δ(''n'', ''F'') by the [[symmetric group]] ''S''<sub>''n''</sub>:
:&Delta;(''n'', ''F'') {{unicode|&#x22C9;}} ''S''<sub>''n''</sub>,
where ''S''<sub>''n''</sub> acts by permuting coordinates and the diagonal matrices Δ(''n'', ''F'') are isomorphic to the ''n''-fold product (''F''<sup>&times;</sup>)<sup>''n''</sup>.
 
To be precise, the generalized permutation matrices are a (faithful) [[linear representation]] of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.
 
===Subgroups===
* The subgroup where all entries are 1 is exactly the [[permutation matrices]], which is isomorphic to the symmetric group.
* The subgroup where all entries are ±1 is the [[signed permutation matrices]], which is the [[hyperoctahedral group]].
* The subgroup where the entries are ''m''th [[roots of unity]] <math>\mu_m</math> is isomorphic to a [[generalized symmetric group]].
* The subgroup of diagonal matrices is abelian, normal, and a maximal abelian subgroup. The quotient group is the symmetric group, and this construction is in fact the [[Weyl group]] of the general linear group: the diagonal matrices are a [[maximal torus]] in the general linear group (and are their own centralizer), the generalized permutation matrices are the normalizer of this torus, and the quotient, <math>N(T)/Z(T) = N(T)/T \cong S_n</math> is the Weyl group.
 
== Properties ==
* If a nonsingular matrix and its inverse are both [[nonnegative matrices]] (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix.
 
== Generalizations ==
One can generalize further by allowing the entries to lie in a ring, rather than in a field. In that case if the non-zero entries are required to be [[unit (ring theory)|units]] in the ring (invertible), one again obtains a group. On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms a [[semigroup]] instead.
 
One may also schematically allow the non-zero entries to lie in a group ''G,'' with the understanding that matrix multiplication will only involve multiplying a single pair of group elements, not "adding" group elements. This is an [[abuse of notation]], since element of matrices being multiplied must allow multiplication and addition, but is suggestive notion for the (formally correct) abstract group <math>G \wr S_n</math> (the wreath product of the group ''G'' by the symmetric group).
 
==Signed permutation group==
{{see|Hyperoctahedral group}}
A '''signed permutation matrix''' is a generalized permutation matrix whose nonzero entries are ±1, and are the integer generalized permutation matrices with integer inverse.
 
===Properties===
* It is the [[Coxeter group]] <math>B_n</math>, and has order <math>2^nn!</math>.
* It is the symmetry group of the [[hypercube]] and (dually) of the [[cross-polytope]].
* Its index 2 subgroup of matrices with determinant 1 is the Coxeter group <math>D_n</math> and is the symmetry group of the [[demihypercube]].
* It is a subgroup of the [[orthogonal group]].
 
==Applications==
===Monomial representations===
{{main|Monomial representation}}
Monomial matrices occur in [[representation theory]] in the context of [[monomial representation]]s. A monomial representation of a group ''G'' is a linear representation ''&rho;'' : ''G'' → GL(''n'', ''F'') of ''G'' (here ''F'' is the defining field of the representation) such that the image ''&rho;''(''G'')  is a subgroup of the group of monomial matrices.
 
==References==
* {{cite book | last=Joyner | first=David | title=Adventures in group theory. Rubik's cube, Merlin's machine, and other mathematical toys | edition=2nd updated and revised | location=Baltimore, MD | publisher=Johns Hopkins University Press | year=2008 | isbn=978-0-8018-9012-3 | zbl=1221.00013 }}
 
[[Category:Matrices]]
[[Category:Permutations]]
[[Category:Sparse matrices]]

Latest revision as of 01:23, 12 January 2015

Oral items provide the herbal components to the bust with the normal digestive procedure while serums and topical creams apply the item straight to the skin. When that isn't the case, then you can expect your imbalanced hormone to be the reason for your enlarged breasts. The Stearate is a very inexpensive synthetic "salt. This boosting of emission is key to the formation of new breast cells and tissue, and the boosting of flow is key to the refining of the estrogen receptor sites which get misty with environmental toxins that mimic estrogen. Natural Breast Enlargement is the total opposite of breast surgery, there's no doctor, knifes or silicone involved when using these natural enhancement methods.

breast enlargement surgery and you may find you are suitable for cheap breast enlargement. Yes, they all have well-endowed breasts, may it be open or artificially enhanced. You have to sleep with it on, go to work with it on, exercise with it on - everything. Breast enhancement cream: There are now breast augmentation creams and gels available which are quite effective. For many women, who want breast enlargement but without the costly, risk prone and painful method of breast enhancing surgery, then the above mentioned natural breast enlargement methods are some of the available options to consider.

When you think about Breast Enlargement, what do you think of first. Dong Quai has been used by the Chinese for more than two thousand years as a strengthener of the heart, lung, spleen, liver and kidney meridians and as a tonic for the blood. ' However, sometimes the pair does not divide and instead sticks together and travels to its place in the dividing cell. Girls don't want Snow White and Cinderella any more. Before undergoing the procedure, the surgeon will decide whether you are fit enough to undergo the procedure.

Though it may be difficult to admit, there are actually a number of women out there who are not satisfied with their current bust size. Many are skeptical about pills, creams, and other gadgets that are advertised to supposedly help them get bigger breasts. Massaging helps to tone these layers and hence provide firmness to the breast. • Likely outcomes of the procedure and any risks or potential complications. Fennel also frequently found in some traditional weight loss ingredients.

Breast enlargement exercises may give you a bigger measurement but cup size and breast size will not be increased. Pushup or padded bras are one option, but there are other seemingly possible ways, as well. Surgical breast implants involves placing saline-filled implants either under the breast tissue or the chest muscle beneath the breasts. We live in a culture where good looks are considered a must, especially for all those who want to move up the ladder. A quick search will return an abundance of results.

Wichai Cherdshewasart, the studies indicated that this sage shows estrogenic and rejuvenate things to the female body especially at the breast, hip, facial skin, body skin, locks and vaginal epithelium. And PMS may subside with intake of Breast Enlargement Pills. The venous drainage of the breast is mainly to the axillary vein, but there is some drainage to the internal thoracic vein and the intercostal veins. These are Silicone and Saline implants, and they come in different shapes, sizes and surface textures. There is no one natural breast enlargement formula that mechanism on all women.

If you have any questions with regards to where by and how to use how to make your breast bigger in a week, you can get hold of us at our own webpage.

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