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		for my $j ( 0 .. $#{$matrix_ref->[$i]} ) {
			$transpose[$j]->[$i] = $matrix_ref->[$i]->[$j];
		}
	}   
	return \@transpose;
}

sub delete_multiple_columns { # if the column elements do not change the final result
   my ( $matrix_ref, $verbose ) = @_;
   my %lower_values;
   my %intersection_columns;
   my $number_of_columns_deleted = 0;	  
   
   for my $i ( 0 .. $#{$matrix_ref} ) {
      for my $j ( 0 .. $#{$matrix_ref->[$i]} ) {
	     $lower_values{ $i }{ $matrix_ref->[$i]->[$j] }{ $j }++;
      }
   } 
   
   # consider N rows < M columns
   # remove the matching columns whose elements are never among the N largest elements in each row
   
   foreach my $index_i ( sort { $a <=> $b } keys %lower_values ){
      my $num_higher_values = 0;
      foreach my $matrix_value ( sort { $b <=> $a } keys %{$lower_values{$index_i}} ){
	     foreach my $index_j ( sort { $b <=> $a } keys %{$lower_values{$index_i}{$matrix_value}} ){	     
		    $intersection_columns{$index_j}++ if ( $num_higher_values++ >= $min_size );
			$number_of_columns_deleted++ if ( defined $intersection_columns{$index_j} and $intersection_columns{$index_j} >= $min_size );
		 }
	  }
   }

   if ( $verbose >= 5 ){
      print "\n";
      for my $i ( 0 .. $#{$matrix_ref} ) {
         print " [";
         for my $j ( 0 .. $#{$matrix_ref->[$i]} ) {
            printf (" %${matrix_spaces}.${decimals}f", $matrix_ref->[$i]->[$j] );
			if ( defined $intersection_columns{$j} and $intersection_columns{$j} == $min_size ){ print "**"; } else{ print "  "; }
         }
         print "]\n";
      }
	  print "\n";
   }	  

   my $idx = 0;  
   for my $i ( 0 .. $#{$matrix_ref} ) {
      for my $j ( 0 .. $#{$matrix_ref->[$i]} ) {
	     undef ( $matrix_ref->[$i]->[$j] ) if ( defined $intersection_columns{$j} and $intersection_columns{$j} >= $min_size );
		 if ( $i == 0 and defined $intersection_columns{$j} and $intersection_columns{$j} >= $min_size ){
		    printf " N = $min_size ; M = $max_size ; j = %2s ; \$intersection_columns{$j} = $intersection_columns{$j} \n", $j if ( $verbose >= 5 );
		 }
		 if ( $i == 0 and ( not defined $intersection_columns{$j} or $intersection_columns{$j} < $min_size ) ){
			$index_correlation{$idx} = $j;
			printf " N = $min_size ; M = $max_size ; j = %2s ; \$index_correlation{$idx} = $index_correlation{$idx} \n", $j if ( $verbose >= 5 );
			$idx++;
		 }
      }
   }
   
	for my $i ( 0 .. $#{$matrix_ref} ) {
		@{$matrix_ref->[$i]} = grep { defined($_) } @{$matrix_ref->[$i]};
	}