Re: all the possible minors of a matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg73775] Re: [mg73755] all the possible minors of a matrix*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 28 Feb 2007 04:31:17 -0500 (EST)*References*: <200702271053.FAA24211@smc.vnet.net>

Mark Novak 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 at http://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 > Recall Minors takes an optional third argument which is a function to use in place of (default) Det. So you could use something like: unsignedCofactors[mat_?MatrixQ, k_] := Module[{len = Length[mat], mn}, (mn = Minors[mat, len - k, permanent]; Reverse[Map[Reverse, mn]]) /; len >= k] where permanent is as you indicated (or can be defined in other ways e.g. using recursion and memo-ization). permanent[mat_?MatrixQ ] := Module[ {t, x}, x = Array[t, Length[mat]]; Coefficient[Apply[Times, mat.x], Apply[Times, x]] ] /; Length[mat] === Length[mat[[1]]] Going from minors to cofactors is discussed in a section of the notebook at http://library.wolfram.com/infocenter/Conferences/325/ Daniel Lichtblau Wolfram Research

**References**:**all the possible minors of a matrix***From:*Mark Novak <mnovak@uchicago.edu>