It is easy enough to manipulate matrices once you have the whole thing in memory (Math::Matrix), but i don't know if there is a clever way of reading by rows and calculating column scores on-the-fly like you want.

Here is one way of at least stripping out the non-zero elements, storing them column-wise and then joining them into one array and printing them out. I would have thought it would be easier to keep the values and indices (@Ai and @Ax) as array refs anyways, it would make it easier to reconstitute the matrix later one if you wanted to?

Anyways, here is my attempt at the problem, as usual, i am sure there are more elegant ways, but KISS always make more sense to me...

#! usr/bin/perl

use strict;
use warnings;
use diagnostics;

# 1.   cumulative count of entries, as matrix is scanned columnwise.
#      Yielding:   Ap = [ 0, 2, 5, 9, 10, 12 ];
# 2. row indices of entries, as matrix is scanned columnwise.
#      Yielding:  Ai = [0, 1, 0,  2, 4, 1,  2, 3, 4, 2, 1, 4 ];
# 3. Non-zero matrix entries, as matrix is scanned columnwise.
#      Yielding: Ax = [2, 3, 3, -1, 4, 4, -3, 1, 2, 2, 6, 1];

my (@Ai, @Ax, @Ap,) = ();
while (<DATA>) {

  chomp;
  my @elements = split /\s+/;
  ## process row-wise
  for (0 .. $#elements){
    ## store non-zero elements by column
    if ($elements[$_] != 0){
      ## store values by column
      push @{ $Ax[$_] }, $elements[$_];
      ## store column index (minus one to make it start at zero)
      push @{ $Ai[$_] }, ($. -1);
      ## increment column totals
      ++$Ap[$_];
    }
  }
}
## make cumulative totals
unshift @Ap, 0;
for (1 .. $#Ap){
  $Ap[$_] +=$Ap[$_-1];
}
## join array refs together
@Ai = map{ @$_ } @Ai;
@Ax = map{ @$_ } @Ax;

print('@Ax  = [', join(" ", @Ax), "]\n");
print('@Ai = [', join(" ", @Ai), "]\n");
print('@Ap = [', join(" ", @Ap), "]\n");

__DATA__
2   3   0   0   0
3   0   4   0   6
0  -1  -3   2   0
0   0   1   0   0
0   4   2   0   1
[download]

Gives :

~ $  perl Desktop/test.pl
@Ax  = [2 3 3 -1 4 4 -3 1 2 2 6 1]
@Ai = [0 1 0 2 4 1 2 3 4 2 1 4]
@Ap = [0 2 5 9 10 12]
[download]

HTH