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MathGroup Archive 1996

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Minors[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4553] Minors[]
  • From: Harald Berndt <haraldb at nature.berkeley.edu>
  • Date: Wed, 7 Aug 1996 04:18:00 -0400
  • Organization: University of California Forest Products Lab
  • Sender: owner-wri-mathgroup at wolfram.com

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
_______________________________________________________________


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