Re: all the possible minors of a matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg73792] Re: all the possible minors of a matrix*From*: "janos" <janostothmeister at gmail.com>*Date*: Wed, 28 Feb 2007 04:40:25 -0500 (EST)*References*: <es13bi$o3s$1@smc.vnet.net>

On Feb 27, 12:06 pm, Mark Novak <mno... at uchicago.edu> wrote: > Hello, > > I'm trying to recode a script into Mathematica and am stuck. > > My problem in short: > I need to calculate all minors of a matrix, but can't figure out a way > to get Mathematica to do more than just a specifically assigned minor at > a time (e.g., the minor produce by removing column 1, row 1). > > My problem in long version: > (I've posted an explanation with example matrices and links to the code > I have written athttp://home.uchicago.edu/~mnovak/mathematicahelp.html) > > The original line is > T:=matrix(n,n,(i,j)->permanent(minor(abs(A),j,i))):evalm(T); > > So, given a matrix A of dimensions n by n, determine the n x n different > minors of the |A| matrix (each minor being of size n-1 by n-1), then > calculate the permanent of each of these minors, and put the resultant > single value into the relevant position of an n by n matrix. That is, > the permanent of the minor produced by removing the ith row and jth > column goes into position (i,j). > > First we need to define how we want the Minor of a matrix to be > calculated (Mathematica's "Minors" function does it in a way that we > don't want.) Second, Mathematica doesn't have a function for calculating > a matrix's permanent, so we need to define that function. (Both of these > I got from searching the the Mathgroup forum.) > > Minor[m_List?MatrixQ, {i_Integer, > j_Integer}]:=Abs[Drop[Transpose[A],{j}]],{i}]] > Permanent[m_List]:=With[{v=Array[x,Length[m]]},Coefficient[Times@@(m.= v),Times@@v]] > > Then the following does work.... > > Minor[Abs[A],{1,3}]//MatrixForm > Permanent[Minor[Abs[A],{i,j}]]/.{i->1,j->1} > > But the problem is that while I can do each of the Minor and Permanent > calculations for specified rows i & columns j of the matrix, I can't > figure out how to do all n x n possible combinations of i and j. > > Any suggestions would be much appreciated. > Thanks! > -mark > > -- > > *************************************** > Dept. of Ecology & Evolution > 1101 E. 57th St., U. of Chicago > Chicago, IL 60637 > Office/Lab: 773-702-4815 > Fax: 773-702-9740http://home.uchicago.edu/~mnovak/ > *************************************** I show you how to calculate all the subdeterminants of the 3x3 matrix A=2E s = Subsets[Range[3]]; tr = Transpose[{s, s}]; A = {{a, b, c}, {d, e, f}, {g, h, j}}; tap = Table[Apply[Part[A, ##] &, tr=E3=80=9Ai=E3=80=9B], {i, 2, 2^3}](*UG= LY*) If you wish you can calculat the detrminants (you would prefer permanents) of these submatrices as Map[Det,tap] 1=2E I hope this helps. 2=2E The UGLY step can be made much more elegantly, would need some time. Janos