I actually did something very similar in here. And tested with 6000*2000 matrix. Here's the code snippet: 
open (INM,"$in_matrix") or die "Unable to open 'in_matrix': $!";
my @row=split (/\t/,readline(INM));
my @colfiles=();
my $max=$#row;
foreach (0 .. $max) {
  open ($colfiles[$_],">$out_matrix.$_") or die "Failed on opening tem
+pfile $_: $!";
  print {$colfiles[$_]} shift (@row);
}

[download]
Message was 
Failed on opening tempfile 1019: Too many open files at ../bin/transposematrix.pl line 62, <INM> line 1.
So is that error from perl or from shell i'm executing the script? I think its from perl, since it tells me the line (correctly) from the script. So it seems to me that i need to do this in 1000 file groups if i do it this way.

Another thing which I though about this morning is combining your original way and idea of splitting the matrix. Now if i split it into 4 pieces, each piece needs to transposed 180 degrees, in which case your original code would do the trick. I figure it should work, even if the split isn't exact (eg. original matrix has 183 rows and 235 columns).

In reply to Re:^4 Binary file handling by Hena
in thread Binary file handling by Hena

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