Summation: Difference between revisions

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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
Any legitimate fat chart can have a ton of information available to help we figure out your ideal fat. First thing to consider is your height because, ideally, we cant anticipate to weigh the same amount because somebody six inches smaller or taller than you!<br><br>For years, it has been assumed which those indicated as "overweight" according to the BMI were at a much higher risk of dying from heart related conditions. But the new research which the Mayo Clinic in Rochester, Minnesota, published indicates something quite different. In their study of 250,000 individuals with heart disease, those with a BMI which indicated overweight status had less chance of dying from heart issues than those with a regular BMI. And persons with a normal BMI were less likely to die than persons with low BMI. And as expected, severely fat persons did have a high incidence of death from heart-related disease.<br><br>When calculating how much weight to lose the most crucial amount to recognize is the Body Fat Percentage. This amount tells we how much of your body is fat, and from there we can calculate how much is muscle, water, etc. For example, in the event you are a 586" female weighing 180 pounds and have a BP of 43%, then 77.4 pounds are fat, and 102.6 pounds are muscle, skeletal mass, and water. We have a BMI of 28.2 - a little over the norm. Then we can figure out that when you want your BF to be 23% - very ideal for a woman - you should lose 36 pounds of fat. The result: a weight of 144 pounds with a BP of 23%, which in turn offers we a BMI of 22.6, well in the regular range.<br><br>Wondering what a fat must be?  Try Mayo Clinics [http://safedietplansforwomen.com/bmi-calculator bmi calculator women].  We input height and weight, the calculator determines your BMI  or body mass index (a formula that estimates body fat). You are able to compare to the norms for under/over and really proper weights. Results is printed.<br><br>Underweight or perhaps a BMI beneath 18.5 may cause reproductive issues inside ladies, heart failure and death. Obesity or a BMI over 30 can lead to heart failure, kidney disease, respiratory illness,an improved risk of cancer plus death.<br><br>In purchase to determine ideal weight for kids and teens (between age group 2-20 years), there is a specific reference tool called the BMI percentile chart. In case, when the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender and age group. According to the chart, a child's body mass index falling inside 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming overweight. On the different hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.<br><br>Note: If you don't recognize your BMI score we can find numerous free online calculators to aid you figure it. One such calculator is found at: wdxcyber.com/bmi. Remember an perfect BMI score is 1 between 19 plus 24.9. An obese BMI score is 1 which falls between 25.9 plus 29.9 plus anything over 30 is considered overweight.
 
:<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]]

Revision as of 14:19, 22 February 2014

Any legitimate fat chart can have a ton of information available to help we figure out your ideal fat. First thing to consider is your height because, ideally, we cant anticipate to weigh the same amount because somebody six inches smaller or taller than you!

For years, it has been assumed which those indicated as "overweight" according to the BMI were at a much higher risk of dying from heart related conditions. But the new research which the Mayo Clinic in Rochester, Minnesota, published indicates something quite different. In their study of 250,000 individuals with heart disease, those with a BMI which indicated overweight status had less chance of dying from heart issues than those with a regular BMI. And persons with a normal BMI were less likely to die than persons with low BMI. And as expected, severely fat persons did have a high incidence of death from heart-related disease.

When calculating how much weight to lose the most crucial amount to recognize is the Body Fat Percentage. This amount tells we how much of your body is fat, and from there we can calculate how much is muscle, water, etc. For example, in the event you are a 586" female weighing 180 pounds and have a BP of 43%, then 77.4 pounds are fat, and 102.6 pounds are muscle, skeletal mass, and water. We have a BMI of 28.2 - a little over the norm. Then we can figure out that when you want your BF to be 23% - very ideal for a woman - you should lose 36 pounds of fat. The result: a weight of 144 pounds with a BP of 23%, which in turn offers we a BMI of 22.6, well in the regular range.

Wondering what a fat must be? Try Mayo Clinics bmi calculator women. We input height and weight, the calculator determines your BMI or body mass index (a formula that estimates body fat). You are able to compare to the norms for under/over and really proper weights. Results is printed.

Underweight or perhaps a BMI beneath 18.5 may cause reproductive issues inside ladies, heart failure and death. Obesity or a BMI over 30 can lead to heart failure, kidney disease, respiratory illness,an improved risk of cancer plus death.

In purchase to determine ideal weight for kids and teens (between age group 2-20 years), there is a specific reference tool called the BMI percentile chart. In case, when the percent value falls at 80, it means the kid is having more fat than 80 % of the kids of the same gender and age group. According to the chart, a child's body mass index falling inside 95 percentile or above indicates he is obese, while those with a BMI above 85 percentile have a risk of becoming overweight. On the different hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.

Note: If you don't recognize your BMI score we can find numerous free online calculators to aid you figure it. One such calculator is found at: wdxcyber.com/bmi. Remember an perfect BMI score is 1 between 19 plus 24.9. An obese BMI score is 1 which falls between 25.9 plus 29.9 plus anything over 30 is considered overweight.

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