Re: List Manipulation

*To*: mathgroup at smc.vnet.net*Subject*: [mg31547] Re: [mg31538] List Manipulation*From*: Tomas Garza <tgarza01 at prodigy.net.mx>*Date*: Sat, 10 Nov 2001 01:19:34 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Recall that "Sign[x] gives -1, 0 or 1 depending on whether x is negative, zero, or positive." (See the on-line Help browser). If In[1]:= listA = {{A, {0, 0, 0, a}}, {B, {b, 0, 0, d}}, {C, {-a, -b, 0, 0}}, {D, {c, 0, 0, -c}}}; there are many possibilities. Consider the following two (here a, b, c, d have not been assigned numerical values; once they have you'll get the appropriate -1, 0, or 1) : In[2]:= ({#1[[1]], Sign[#1[[2]]]} & ) /@ listA Out[2]= {{A, {0, 0, 0, Sign[a]}}, {B, {Sign[b], 0, 0, Sign[d]}}, {C, {-Sign[a], -Sign[b], 0, 0}}, {D, {Sign[c], 0, 0, -Sign[c]}}} In[3]:= listA /. {x_, y_List} -> {x, Sign[y]} Out[3]= {{A, {0, 0, 0, Sign[a]}}, {B, {Sign[b], 0, 0, Sign[d]}}, {C, {-Sign[a], -Sign[b], 0, 0}}, {D, {Sign[c], 0, 0, -Sign[c]}}} These work because Sign has the attribute of listability (q.v.) and so it operates on each of the elements of the inner list. Tomas Garza Mexico City ----- Original Message ----- From: "Jonathan Woodward" <woodward at chem.ufl.edu> To: mathgroup at smc.vnet.net Subject: [mg31547] [mg31538] List Manipulation > I am a relatively new user to Mathematica with virutally no > programming experience and need help with a problem: > > Given the following hypothetical eigensystem, a "list of lists" which > has the eigenvalues (A,B,C,D) associated with their corresponding > eigenvectors ({0,0,0,a},{b,0,0,d},{-a,-b,0,0},{c,0,0,-c}): > > {{A,{0,0,0,a}},{B,{b,0,0,d}},{C,{-a,-b,0,0}},{D,{c,0,0,-c}}} > > where the list contains zeros, symbolic expressions, and numbers. > The actual system I have is the eigensystem of a 32x32 symbolic matrix > where the vector components seem to take up hundreds of pages and > are therefore almost useless to me. However, I am interested in the > position of the zero and nonzero components only, not their actual > values. > > So what I want to do is transform the list into another more > useful list in the following way: > > I do not want to change the eigenvalues but want to convert all > eigenvector > components in such a way that I have a list of zeros, ones, and > negative ones. > In other words, divide each eigenvector component by its absolute > value, except for the zeros, to create a new list that might now look > like: > > {{A,{0,0,0,1}},{B,{1,0,0,1}},{C,{-1,-1,0,0}},{D,{1,0,0,-1}}} > > This way I can greatly simplify my problem while keeping the position > of the > zero and nonzero elements of the components unchanged. > > How would I write a code in Mathematica to accomplish this? In > particular, > how would do I tell the program to scan through this list, doing > nothing > to the eigenvalues, but look through the eigenvectors, check to see if > they > are nonzero (if zero, do nothing) and divide each nonzero component by > its > absolute value, and return a new list. I don't want to break the list > apart > and operate just on the vector components themselves because I want to > preserve the eigenvalue-eigenvector association. Also, I need to be > able to tell > the program that symbols of the type {x} are positive and {-x} are > negative > otherwise I might have a list returned like: > > {{A,{0,0,0,a/Abs[a]}},{B,{b/Abs[b],0,0,d/Abs[d]}},{C,{-a/Abs[a],-b/Abs[b] ,0, 0}},{D,{c/Abs[c],0,0,-c/Abs[c]}}} > > which is not simplified to what I need. > > Any help would be greatly appreciated. > > Thanks > > Jonathan Woodward >