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Minors[]
Posted:
Aug 1, 1996 2:05 AM


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 builtin 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 columnexpanding them appropriately. It would have been real nice to use the Minors[] function, if only it would return the subdeterminants in the expected positions!
 _______________________________________________________________ Harald Berndt University of California Research Specialist Forest Products Laboratory Phone: 5102154224 FAX:5102154299 _______________________________________________________________



