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Re: Re: Minors

  • To: mathgroup at
  • Subject: [mg53225] Re: [mg53129] Re: [mg52844] Minors
  • From: Garry Helzer <gah at>
  • Date: Sat, 1 Jan 2005 02:33:45 -0500 (EST)
  • References: <> <> <>
  • Sender: owner-wri-mathgroup at

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 

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.


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> 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
>> Garry Helzer
>> gah at
> -- 
> DrBob at
Garry Helzer
gah at

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