permanent

permanent

Definition of the permanent

Let

M

be an

n

×

n

matrix over a field (or even a

commutative ring

)

𝖨𝖥

, and

M

i

⁢

j

its entries (with

i

ranging over a set

I

and

j

over a set

J

, each of

n

elements

).

The

permanent

of

M

, denoted by

per

M

, is given by

per

M

=

def

∑

π

∈

S

∏

i

∈

I

M

i

⁢

π

⁢

(

i

)

where

S

is the set of all

n

!

bijections

π

:

I

→

J

. Usually

I

and

J

are identified with each other (and with the set of the first

n

natural numbers

, traditionally skipping 0) so that

S

consists of the elements of the

symmetric group

S

n

, the group of

permutations

of

n

objects.

In words: we want

products

of each time

n

matrix elements

chosen
such that there’s one from each row

i

and also one from each column

j

.
There are

n

!

ways to pick those elements (for any permutation of the
column indices relative to the row indices take the elements that end
up in

diagonal

position). The permanent is the sum of those

n

!

products.
E.g. (

n

=

3

)

@ O O O @ O O O @ @ O O O @ O O O @

O @ O + O O @ + @ O O + O O @ + @ O O + O @ O

O O @ @ O O O @ O O @ O O O @ @ O O

Comparison with the determinant

It is closely related to one of the ways to define the

determinant

,

det

⁡

M

=

def

∑

π

∈

S

I

±

∏

i

∈

I

M

i

⁢

π

⁢

(

i

)

where

S

I

is the

symmetry group

of

I

(

isomorphic

to

S

n

); the sign is

+

for

even permutations

and

-

for odd ones.
E.g. (

n

=

3

)

@ O O O @ O O O @ @ O O O @ O O O @

O @ O + O O @ + @ O O - O O @ - @ O O - O @ O

O O @ @ O O O @ O O @ O O O @ @ O O

Note two important

differences

though.

•

While the determinant enjoys the property that

(

det

⁡

A

)

⁢

(

det

⁡

B

)

=

det

⁡

(

A

⁢

B

)

, the permanent has no such nice

arithmetic

properties.

•

In the definition of the determinant, we must have

I

=

J

i.e. we must
have a particular way in mind of

matching

rows to columns. A matrix
to transform from one basis to another would be an example where this
match is arbitrary. Different conventions which row goes with which
column give two different values (one

-

1

times the other) for the
determinant. By contrast, the permanent is well defined for any

I

and

J

.

In the

representation theory

of groups (where the field

𝖨𝖥

is

ı

ı

⁢

𝖢

) determinant and permanent are special cases of the

immanent

(there is an immanent for every

character

of

S

n

).

Some properties of the permanent

These follow immediately from the definition:

•

The permanent is “

multilinear

” in the rows and columns (i.e. linear in every one of them).

•

It is “

homogeneous of degree

n

”, i.e.

per

(

k

⁢

A

)

=

k

n

⁢

per

(

A

)

,
where

k

is a scalar (i.e. element of

𝖨𝖥

).

•

When

P

and

Q

are

permutation matrices

,

per

(

P

⁢

A

⁢

Q

)

=

per

(

P

⁢

A

)

=

per

(

A

⁢

Q

)

=

per

(

A

)

.

•

per

(

A

⊤

)

=

per

(

A

)

.

•

per

(

A

⁢

0

0

⁢

D

)

=

per

(

A

)

⁢

per

(

D

)

.

•

If

M

only has nonnegative entries, then

per

M

⩾

det

⁡

M

Equality is attained

–

(with value 0) when the products (of entries, one from each row
and from each column) that contribute to det and per are all zero.
This happens whenever a whole column or row is zero, but also for

instance

in

(

12

0

0

13

0

0

14

15

16

)

–

when the permutations used are all even (which implies only
one is used). This happens in a

diagonal matrix

, but also
for instance in

(

0

1

0

0

1

0

0

0

0

0

0

1

0

0

1

0

)

Permanents find application in

combinatorics

, e.g. in enumerating

Latin
squares

.

See also Van der Waerden’s permanent conjecture (now a

theorem

) for

doubly
stochastic matrices

(where

𝖨𝖥

is

𝖨𝖱

).

Note: some authors define something for non-square matrices they also call “permanent”.

Title

permanent

Canonical name

Permanent

Date of creation

2013-03-22 14:05:39

Last modified on

2013-03-22 14:05:39

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

12

Author

marijke (8873)

Entry type

Definition

Classification

msc 15A15

Related topic

Immanent

Related topic

Determinant2

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