```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}])//TableFormOut[1]//TableForm=a     a     a 11    12    13a     a     a 21    22    23a     a     a 31    32    33... then calculate it's minors using Mma's built-in function:In[2]:=(tmMM = Minors[tm, 2])//TableFormOut[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  33OK, 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 structureIn[3]:=(minTab = Reverse[Transpose[Reverse[Transpose[Table[	Subscripted[		m[ToString[i]<>ToString[j]]	], {i, 3}, {j, 3} ]]]]])//TableFormOut[3]//TableForm=m     m     m 33    32    31m     m     m 23    22    21m     m     m 13    12    11Why 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 CaliforniaResearch Specialist                  Forest Products LaboratoryPhone: 510-215-4224                            FAX:510-215-4299_______________________________________________________________
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