For a given matrix, return the determinant and the permanent of the matrix.

The determinant is given by

 



det
(
A
)
=

∑

σ


sgn
⁡
(
σ
)

∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \det(A)=\sum _{\sigma }\operatorname {sgn}(\sigma )\prod _{i=1}^{n}M_{i,\sigma _{i}}}


while the permanent is given by

 



perm
⁡
(
A
)
=

∑

σ



∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \operatorname {perm} (A)=\sum _{\sigma }\prod _{i=1}^{n}M_{i,\sigma _{i}}}


In both cases the sum is over the permutations 



σ


{\displaystyle \sigma }

 of the permutations of 1, 2, ..., n. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see parity of a permutation.)

More efficient algorithms for the determinant are known: LU decomposition, see for example wp:LU decomposition#Computing the determinant. Efficient methods for calculating the permanent are not known.



Related task

 Permutations by swapping


package require math::linearalgebra
package require struct::list
 
proc permanent {matrix} {
    for {set plist {};set i 0} {$i<[llength $matrix]} {incr i} {
	lappend plist $i
    }
    foreach p [::struct::list permutations $plist] {
	foreach i $plist j $p {
	    lappend prod [lindex $matrix $i $j]
	}
	lappend sum [::tcl::mathop::* {*}$prod[set prod {}]]
    }
    return [::tcl::mathop::+ {*}$sum]
}