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{{Expert-subject|Mathematics|date=March 2011}}
In [[mathematics]], a '''balanced matrix''' ''B'' is a 0-1 matrix that does not contain any square submatrix of odd order having row and column sum equal to 2.


Balanced matrices are important in linear programs such as the [[set partitioning problem]], as they are naturally integer. 0-1 [[Totally unimodular]] matrices are a subset of balanced matrices, and balanced matrices are a subset of [[perfect matrix|perfect matrices]], therefore any matrix that is totally unimodular is also balanced, however a balanced matrix may not necessarily be [[totally unimodular]].


The following matrix is a 3 order 2-cycle forbidden submatrix:
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:<math>\begin{bmatrix}
1 & 0 & 1\\
1 & 1 & 0\\
0 & 1 & 1\\
\end{bmatrix}</math>
 
The following matrix is a balanced matrix as it does not contain the above nor any other odd order 2-cycle submatrix:
:<math>B=\begin{bmatrix}
1 & 1 & 1 & 1\\
1 & 1 & 0 & 0\\
1 & 0 & 1 & 0\\
1 & 0 & 0 & 1\\
\end{bmatrix}</math>
 
The following matrix is a 5 order forbidden submatrix:
:<math>\begin{bmatrix}
1 & 0 & 0 & 0 & 1\\
1 & 1 & 0 & 0 & 0\\
0 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 1 & 1\\
\end{bmatrix}</math>
 
==Subsequence count==
{{citation needed|date=August 2012}}An alternative method of identifying a balanced matrix that is also a zero-one matrix is through the subsequence count, where the subsequence count ''SC'' of any row s of matrix ''A'' is
 
:'''SC''' = |{''t'' | [''a''<sub>''sj''</sub>&nbsp;=&nbsp;1, ''a''<sub>''ij''</sub>&nbsp;=&nbsp;0 for ''s''&nbsp;<&nbsp;''i''&nbsp;<&nbsp;''t'', ''a''<sub>''tj''</sub>&nbsp;=&nbsp;1], ''j''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n''}|
If a matrix ''A'' has SC(''s'')&nbsp;≤&nbsp;1 for all rows ''s''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''m'', then ''A'' has a unique subsequence, is totally unimodular<ref>Ryan &amp; Falkner 1988</ref> and therefore also balanced. Note that this condition is sufficient but not necessary for ''A'' to be balanced.
 
== Notes ==
{{Reflist}}
 
== References ==
*{{Citation
  | last = Berge
  | first = C.
  | year = 1972
  | title = Balanced Matrices
  | publisher = Centre National de Recherche Scientifique
  | location = Paris, France}}
 
*{{Citation
  | last1 = Ryan | first1 = D. M.
  | last2 = Falkner | first2 = J. C.
  | year = 1988
  | title = On the integer properties of scheduling set partitioning models
  | doi = 10.1016/0377-2217(88)90233-0
  | journal = European Journal of Operational Research
  | volume = 35 | issue = 3 | pages = 442–456}}
 
{{DEFAULTSORT:Balanced Matrix}}
[[Category:Matrices]]

Latest revision as of 00:42, 12 December 2014


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On the road to the airport find your way to Lake Caburgua and then Trancura River. You want to be pedaling as hard as you can while other riders right next to you are doing the exact same thing. An alternative to Clipless pedals are Mountain Bike shoes with Shimano Pedaling Dynamics (SPD). Cost shouldn't be a factor as a good helmet can be acquired for around $25-$45. Before heading to your local bike store to see what Santa Cruz bikes are available, it is advisable to visit the Santa Cruz Bicycles website to see all of their latest bikes.

They have great suspension that fits the need of both amateur and professional bikers. You always have to be ready while riding your bike. The tube sits next to your mouth so you can sip as you ride. It is pretty incredible how much of a significant difference Haro mountain bike can have, of course you have to think about certain factors. If you really want a quality bike which will hold up to the rough terrain with out costing a fortune, this is a great choice.

Consequently, they became like villains in the eyes of the NBA fans outside Miami. He just stood there looking at the view from the top-his view and perspective changed by a few seconds and a climb up a little mountain. Bike Comfort If you are at a height or have a leg length that is somewhere in the middle of 2 bike frame sizes, trying several of them to see what feels more comfortable is the best option. With the tips given, I am sure that you and your new bike will work together for that ultimate mountain biking adventure. The biker will encounter numerous obstacles along the way which will make his trajectory an extreme adventure.

Looked like one or two other people beat me to the punch. Instability: You are much more likely to lose your balance while on a big bike. Are you going to be going up and down hills, across flat terrain, on cement, or through rough and rocky areas. The Red loop has a lot of climbing and some very tight turns. They will provide you with a honest price, yet be ready to pay in between $650 and $4,000 on dual suspensions and in between $470 and $670 for hardtails.

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