Jump to content

  • Log In with Google      Sign In   
  • Create Account

We're offering banner ads on our site from just $5!

1. Details HERE. 2. GDNet+ Subscriptions HERE. 3. Ad upload HERE.


A subtly in property #1 in identifying a ref or rref matrix


Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.

  • You cannot reply to this topic
6 replies to this topic

#1 warnexus   Prime Members   -  Reputation: 1472

Like
0Likes
Like

Posted 22 February 2014 - 09:28 PM

subtlety in property #1 in identifying a ref or rref matrix

 

My linear algebra book says for property #1 as follows:

 

1) If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading 1.

 

Then I came up with my own matrix in my mind and started questioning property #1.

 

2 1 5 1

0 0 1 0

 

Would the number 2 in the first row of this matrix be the "leading 1" ...after all the textbook did say "first nonzero number"?


Edited by warnexus, 22 February 2014 - 09:32 PM.


Sponsor:

#2 Bacterius   Crossbones+   -  Reputation: 9068

Like
0Likes
Like

Posted 22 February 2014 - 09:49 PM

No, that is because that matrix is not in reduced row echelon form. You need to divide the first row by 2 and then subtract 2.5 times the second row from the first row to make it row reduced, at which point you get:

1 0.5   0 0.5
0   0   1   0

Which is rref as it is in row echelon form, has leading coefficient 1 in every row, and every leading coefficient is the only nonzero entry in its coumn.

 

The definition seems to not be universal, though, according to Wikipedia some authors let a row echelon form matrix have leading coefficients that aren't 1, while others require them to be 1. Either way however a reduced row echelon form matrix (like the one above in my post) must have leading coefficients of 1 as that representation is unique.


The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

 

- Pessimal Algorithms and Simplexity Analysis


#3 warnexus   Prime Members   -  Reputation: 1472

Like
0Likes
Like

Posted 23 February 2014 - 04:19 PM


No, that is because that matrix is not in reduced row echelon form.

 

Oh now I get it. You have to reduced it the form before you can apply the 4 properties as a confirmation. That does make more sense. Thanks!



#4 Paradigm Shifter   Crossbones+   -  Reputation: 5411

Like
0Likes
Like

Posted 23 February 2014 - 04:30 PM

Well rref doesn't come up much in game dev anyway, since the matrices are so small.


"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley

#5 warnexus   Prime Members   -  Reputation: 1472

Like
0Likes
Like

Posted 23 February 2014 - 06:47 PM

Well rref doesn't come up much in game dev anyway, since the matrices are so small.

Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.



#6 Bacterius   Crossbones+   -  Reputation: 9068

Like
0Likes
Like

Posted 23 February 2014 - 06:51 PM

 

Well rref doesn't come up much in game dev anyway, since the matrices are so small.

Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.

 

 

4x4 matrices are used all over advanced physics, especially relativity where you need it pretty much for the same reasons as computer graphics (a 3D affine space). Higher dimension matrices are used in various branches of applied statistics, etc... so, no, matrices are not only for CG, they have a wide range of uses across many disciplines.


The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

 

- Pessimal Algorithms and Simplexity Analysis


#7 warnexus   Prime Members   -  Reputation: 1472

Like
0Likes
Like

Posted 23 February 2014 - 10:36 PM

 

 

Well rref doesn't come up much in game dev anyway, since the matrices are so small.

Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.

 

 

4x4 matrices are used all over advanced physics, especially relativity where you need it pretty much for the same reasons as computer graphics (a 3D affine space). Higher dimension matrices are used in various branches of applied statistics, etc... so, no, matrices are not only for CG, they have a wide range of uses across many disciplines.

 

Interesting. Thanks.






Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.



PARTNERS