Date: Aug 1, 1996 2:05 AM
Author: Harald Berndt
Subject: Minors[]

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
_______________________________________________________________