Here is how you can multiply two arrays in the form of matrixes using a bit of matrix algebra (M*M).
By calling the function multiplyMatrix, you will be multiplying two sparse matrixes (zeros need not be included in the array for the operation to be performed).

<?php
$M = array(
0=>array(1=>1,4=>1),
1=>array(2=>1,3=>1),
3=>array(1=>1),
4=>array(5=>1),
5=>array(6=>1)
);

$M1 = multiplyMatrix($M, $M); //multiplying $M by itself

echo '<pre>';print_r($M1);echo '</pre>';

function multiplyMatrix($M1, $M2)
    {
#Helena F Deus, Oct 06, 2008
##Multiply two matrixes; $M1 and $M2 can be sparse matrixes, the indexes on both should match
        if(is_file($M1)) {$matrix1 = unserialize(file_get_contents($M1));}
        else $matrix1 = $M1;
        
            
        #transpose M2
        $M2t = transpose($M2);
        
        foreach ($M2t as $row=>$tmp) {
            ##sum the result of the value in the col multiplied by the value in the vector on the corresponding row
                
                foreach ($M1 as $row1=>$tmp1) {
                    
                    $multiply[$row1] = array_rproduct($tmp,$tmp1);
                    
                    if(!$multiply[$row1]){
                          exit;
                        }
                }
                
                foreach ($multiply as $row1=>$vals) {
                    
                    $sum[$row][$row1]=array_sum($vals);
                }
                
        }
    
    $r=transpose($sum);
    
    return ($r);
    }

function transpose($M)
{
foreach ($M as $row=>$cols) {
            
            foreach ($cols as $col=>$value) {
                 if($value)
                 $Mt[$col][$row]=$value;
            }
        }
        ksort($Mt);
        
return ($Mt);            
}

function array_rproduct($a1, $a2)
{
    
    
    foreach ($a1 as $line=>$cols) {
        $a3[$line] = $a1[$line]*$a2[$line];
        foreach ($a2 as $line2=>$cols2) {
            $a3[$line2] = $a1[$line2]*$a2[$line2];
        }
    }    
    ksort($a3);
    
    
    return ($a3);
    
    
}

?>
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