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#1 2013-12-17 02:36:00

Reuel
Member
Registered: 2010-11-28
Posts: 178

Transformation Matrix

Hello.

How does one go about transforming an arbitrary matrix, say, one of the form

to the form

where the only change are the Greek components?

I tried just working out a matrix P on the spot to multiply M by to get N but I didn't get what I wanted at all. I have had a little experience with matrix transformations but I don't know how to derive the transformation itself. If someone could give me the matrix and explain how you got it so I understand, I would appreciate it.

Thank you for your help.

Last edited by Reuel (2013-12-17 02:36:24)

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#2 2013-12-17 02:49:44

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

This is just an observation but it does appear that

are the only values they can take.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2013-12-17 02:59:44

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

Why isn't it enough to just make the transformation matrix

which is what I originally tried? Wouldn't you just multiply through and get N? Or is multiplying 3x3 matrices that different from 2x2?

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#4 2013-12-17 03:10:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

M X P yields:


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2013-12-17 03:11:52

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Transformation Matrix

Multiplying 3x3 matrices is pretty much the same as multiplying 2x2 matrices.

Where does this come from?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#6 2013-12-17 03:12:12

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

Yeah, I got that too.

So the answer is there is no transformation that will work?

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#7 2013-12-17 03:16:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

There are an infinite number of P's that will work.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#8 2013-12-17 03:18:20

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

Such as?

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#9 2013-12-17 03:29:47

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

That is where I am having a problem. Each solution insists that alpha and beta are 1 and only 1. Therefore I can not come up with an answer that involves them.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2013-12-17 03:37:37

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Transformation Matrix

anonimnystefy wrote:

Where does this come from?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#11 2013-12-17 03:51:48

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

The problem I introduced stemmed from experimentation with representing that hyperbolic function I keep bringing up in matrix form. As bob bundy implied, the function

may be rewritten as

which, according to wikipedia, is of the form

where 2B=1, 2D=-a/c, and 2E=b/c, and everything else is zero. Therefore we can state the original f(x)=y in matrix form as

where

and

In coming here I just used Greek letters to make the problem more simple rather than using fractions for every term. I left the 1/2's in case they held significance.


So then the problem became finding a matrix P that would transform a more basic equation y=x into the equation f(x) given at the start. The function y=x has the matrix

but that didn't work at all which is why I added in the two extra 1/2's in to M because if those terms were left as 0s then no matter what I multiplied by it, I would still get 0. And so, essentially, my original question was to ask for a transformation matrix P that would transform R in to M; that is, to go from

to

using matrices and a matrix transformation.

Last edited by Reuel (2013-12-17 03:54:53)

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#12 2013-12-17 03:53:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

You may have distilled the problem down too much. The system of linear equations your distilled version produces tend to obliterate alpha and beta.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2013-12-17 04:01:38

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

Quite possible. I was just trying to simplify the model for the sake of those reading it but I may have messed it up in doing so.


If simplification complicated the issue, the function f(x) might even be able to expand with some manipulation. For instance letting c=a-b or something to eliminate c while adding in additional a and b terms. Or not. Actually, that might make it worse.

Last edited by Reuel (2013-12-17 04:04:19)

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#14 2013-12-17 04:18:01

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

I am pretty sure that the problem as posted in post #1 will always yield a value of alpha and beta equaling 1.

This may not be bad just set those fractions equal to 1.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2013-12-17 04:32:05

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

You mean rewrite M as

and then N as

?

That makes it look cleaner. I still don't know how to find P to go from M to N, but I am experimenting with it on paper.

Last edited by Reuel (2013-12-17 04:41:15)

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#16 2013-12-17 04:36:50

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

Why change M, it does not have alpha or beta in it?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2013-12-17 04:39:38

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

Consistency. And it still works in that it still leads to y=x.

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#18 2013-12-17 04:41:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

Okay, I am big fan of doing what works. Play with it yourself and if you can not get a solution I will try.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2013-12-17 04:43:11

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

What works and whatever is most simple. smile

I'll keep working on it and come back either with a solution or more whining.

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#20 2013-12-17 04:47:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

Math makes everybody whine. Good luck.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#21 2013-12-17 09:49:55

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Transformation Matrix

I did it but it's more typing than I have time for right now. You have to multiply M by a matrix with 9 unknowns so that you end up with a system of nine equations with 11 unknowns and then solve it for those unknowns entirely in terms of a and b only. It was quite a bit of substitution and algebraic manipulation and the resulting transformation matrix is enormous.

Thanks for your input!

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#22 2013-12-17 10:27:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Transformation Matrix

Hi;

Glad you did it. Come back if anything goes wrong. Good luck.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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