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MathGroup Archive 2005

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Re: Re: Minors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53225] Re: [mg53129] Re: [mg52844] Minors
  • From: Garry Helzer <gah at math.umd.edu>
  • Date: Sat, 1 Jan 2005 02:33:45 -0500 (EST)
  • References: <200412141100.GAA24699@smc.vnet.net> <200412241059.FAA05862@smc.vnet.net> <opsjiwzrfciz9bcq@monster.ma.dl.cox.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Physicists have uses for wedge products in the Clifford algebra context 
but I am not familiar with these applications so I will not comment 
further on them.

If the 4-tuples p,q,r are homogeneous coordinate vectors for points in 
space then the 6-tuple p^q represents the line through p and q and the 
4-tuple p^q^r represents the plane through the three points. This 
representation is unique up to a constant multiple. These 
representations provide an efficient, if underused, tool for 
manipulating lines and planes in graphics programs. (Compare 
representing a line by a 6-tuple unique up to a constant multiple with 
representing it by a pair of linear equations unique up to the reduced 
echelon form of the corresponding matrix.)

If A is the 4-by-4 matrix of a projective transformation on points, 
then the 6-by-6 matrix Minors[A,2] transforms lines and the 4-by-4 
matrix Minors[A,3] transforms planes.

For a specific example (2D) look at the background graphic on the web 
page

http://www.math.umd.edu/~gah/

This is a particular example of a line roulette: a curve defined as the 
envelope of a family of lines generated by a line fixed to one curve as 
that curve rolls on a second curve. Here the orange line is fixed to 
the large circle which rolls on the small circle.  If we look at a 
point in a fixed relation to the large circle the usual trig arguments 
can be used to find a formula for the new position of the point after a 
rotation through the angle t. This rotation has a 3-by-3 matrix A(t) in 
homogeneous coordinates and the 3-by-3 matrix B(t)=Minors[A(t),2] 
applied to the 3-tuple representation of the orange line gives its 
image under the rotation.  This is how the graphic was generated. For 
details follow the link to the lecture notes.

--Garry

Dec 24, 2004, at 9:46 AM, DrBob wrote:

> Convenient for what?
>
> Bobby
>
> On Fri, 24 Dec 2004 05:59:24 -0500 (EST), Garry Helzer 
> <gah at math.umd.edu> wrote:
>
>> If A is the matrix of a linear transformation T on R^n, then
>> Minors[A,k] is the matrix of the induced transformation on the k-th
>> exterior product--that is the induced transformation on k-vectors
>> defined by (using ^ for the wedge product)
>>
>> T(a1^a2^ . . . ^ak)=T(a1)^T(a2)^ . . . ^T(ak)
>>
>> I don't know if this is the actual reason, but it is certainly
>> convenient.
>>
>> On Dec 14, 2004, at 3:00 AM, Robert M. Mazo wrote:
>>
>>> The Minors command gives, as the (i,j) minor af an nxn matrix, what
>>> ordinary mathematical notation calls the (n-i+1,n-j+1) minor .  I 
>>> know
>>> how to work around this.  It is explained on pg. 1195 of The
>>> Mathematica Book (version 4).  My question here is, why did the
>>> programmers of Mathematica define Minors this unconventional way?
>>> They usually had a good reason for their programming quirks, but I
>>> can't think of a reason for this one.  Can anyone enlighten me?
>>>
>>> 	Robert Mazo
>>> 	mazo at uoregon.edu
>>>
>>>
>> Garry Helzer
>> gah at math.umd.edu
>>
>>
>>
>>
>
>
>
> -- 
> DrBob at bigfoot.com
> www.eclecticdreams.net
>
>
Garry Helzer
gah at math.umd.edu


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