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Re: Minors[]
Posted:
Aug 2, 1996 2:04 AM


In article <4tphg4$kl3@dragonfly.wolfram.com> Harald Berndt <haraldb@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 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 > _______________________________________________________________ >
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@wolfram.com



