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