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MathGroup Archive 1996

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Re: Minors[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4568] Re: Minors[]
  • From: danl (Daniel Lichtblau)
  • Date: Fri, 16 Aug 1996 05:15:02 -0400
  • Organization: Wolfram Research, Inc.
  • Sender: owner-wri-mathgroup at wolfram.com

In article <4tphg4$kl3 at dragonfly.wolfram.com> Harald Berndt  
<haraldb at nature.berkeley.edu> writes:
> Hi, All:
> 
> I would like to know the reasoning behind the arrangement of output in 
> Mma's Minors[] function. According to the definition (see, e.g., James 
> and James: Mathematics Dictonary) the minor of an element in a 
> determinant is the determinant, of next lower order, obtained by 
> striking out the row and column in which the element lies.
> 
> Let's look at an example of Mma's treatment:
> 
> Define a 3x3 matrix ...
> 
> In[1]:=
> (tm = Table[ Subscripted[ "a"[ToString[i]<>ToString[j]] ],
> 	{i, 3}, {j, 3}
> ])//TableForm
> Out[1]//TableForm=
> a     a     a
>  11    12    13
> 
> a     a     a
>  21    22    23
> 
> a     a     a
>  31    32    33
> 
> ... then calculate it's minors using Mma's built-in function:
> 
> In[2]:=
> (tmMM = Minors[tm, 2])//TableForm
> Out[2]//TableForm=
> -(a   a  ) + a   a     -(a   a  ) + a   a     -(a   a  ) + a   a
>    12  21     11  22      13  21     11  23      13  22     12  23
> 
> -(a   a  ) + a   a     -(a   a  ) + a   a     -(a   a  ) + a   a
>    12  31     11  32      13  31     11  33      13  32     12  33
> 
> -(a   a  ) + a   a     -(a   a  ) + a   a     -(a   a  ) + a   a
>    22  31     21  32      23  31     21  33      23  32     22  33
> 
> OK, I got all the minors, but at position [[1, 1]], I have the minor 
> associated with a33, not that associated with a11. Call the minors 
> derived according to the definition mij, i.e., mij is the minor 
> obtained by striking out row i and column j, then, the Mma resullt of 
> Minors[] has the structure
> 
> In[3]:=
> (minTab = Reverse[Transpose[Reverse[Transpose[Table[
> 	Subscripted[
> 		m[ToString[i]<>ToString[j]]
> 	], {i, 3}, {j, 3} ]]]]])//TableForm
> Out[3]//TableForm=
> m     m     m
>  33    32    31
> 
> m     m     m
>  23    22    21
> 
> m     m     m
>  13    12    11
> 
> Why was this done? I recently had to deal with the determinants of 4x4 
> symbolic matrices, which I found could be simplified significantly 
> after row- or column-expanding them appropriately. It would have been 
> real nice to use the Minors[] function, if only it would return the 
> sub-determinants in the expected positions!
> 
> -- 
> _______________________________________________________________
> Harald Berndt                          University of California
> Research Specialist                  Forest Products Laboratory
> Phone: 510-215-4224                            FAX:510-215-4299
> _______________________________________________________________
> 

Minors orders its result lexicographically according to the rows and  
columns used in computing them. Thus for 2x2 minors of a 4x4 matrix the  
first row will have minors formed from rows (1,2). The cols of that first  
row will be, in order, minors formed from cols (1,2), cols (1,3), cols  
(1,4), cols (2,3), cols (2,4), and cols (3,4). 

The ordering you have in mind could be achieved by reverse lex ordering  
(or by lex ordering according to the rows and cols omitted). Possibly in  
future we should add some option to control this.

Daniel Lichtblau
Wolfram Research, Inc.
danl at wolfram.com


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