Dear oshalla,
Thank you so much for your reply. It is truly useful.

I also tried to modify your code to do multiplication of transpose of the sparse matrix. With the following code:

use Data::Dumper;
my @realval = (
    0.994704478,
    0.989459074,
    0.994717023,
    1.000000000, 
    1.000000000,
    0.002647761,
    0.005282977, 
    0.000882587, 
    0.005270463, 
    0.000882587, 
    0.002635231, 
    0.000882587,
    0.002635231,);

my @row_index = qw( 1 2 3 4 5 1 3 1 2 1 2 1 2);
my @col_index =  qw(1 2 3 4 5 5 3 2 1 3 3 4 4);
my @mat_dim  = (5,5); # M always equal to N

my @trans_r = (0) x scalar(@p);

for my $j (0..$#row_index) {
 $trans_r[$col_index[$i]-1] += $realval[$i] * $p[$row_index[$i]-1] ;
}

print Dumper \@trans_r;
[download]

Not sure why it doesn't give this desired result:

@result = (0.3989359, 0.1982448, 0.2008801, 0.2008801, 0.2010591);
[download]

Truly need your expert advice on this. Thanks and hope to hear from you again.