proof of Birkhoff-von Neumann theorem

proof of Birkhoff-von Neumann theorem

First, we prove the following lemma:

Lemma:

A

convex combination

of

doubly stochastic matrices

is

doubly stochastic

.

Proof:

Let

{

A

i

}

i

=

1

m

be a collection of

n

×

n

doubly-stochastic matrices, and suppose

{

λ

i

}

i

=

1

m

is a collection of scalars satisfying

∑

i

=

1

m

λ

i

=

1

and

λ

i

≥

0

for each

i

=

1

,

…

,

m

.
We claim that

A

=

∑

i

=

1

m

λ

i

⁢

A

i

is doubly stochastic.

Take any

i

∈

{

1

,

…

,

m

}

. Since

A

i

is doubly stochastic, each of its rows and columns sum to 1. Thus each of the rows and columns of

λ

i

⁢

A

i

sum to

λ

i

.

By the definition of elementwise summation, given matrices

N

=

M

1

+

M

2

, the sum of the entries in the

i

th column of

N

is clearly the sum of the sums of entries of the

i

th columns of

M

1

and

M

2

respectively. A

similar

result holds for the

j

th row.

Hence the sum of the entries in the

i

th column of

A

is the sum of the sums of entries of the

i

th columns of

λ

k

⁢

A

k

for each

i

, that is,

∑

k

=

1

m

λ

k

=

1

. The sum of entries of the

j

th row of

A

is the same. Hence

A

is doubly stochastic.

Observe that since a

permutation matrix

has a single nonzero entry, equal to 1, in each row and column, so the sum of entries in any row or column must be 1. So a permutation matrix is doubly stochastic, and on applying the lemma, we see that a convex combination of permutation matrices is doubly stochastic.
This provides one

direction

of our proof; we now prove the more difficult direction: suppose

B

is doubly-stochastic. Define a

weighted graph

G

=

(

V

,

E

)

with

vertex set

V

=

{

r

1

,

…

,

r

n

,

c

1

,

…

,

c

n

}

,

edge set

E

, where

e

i

⁢

j

=

(

r

i

,

c

i

)

∈

E

if

B

i

⁢

j

≠

0

, and edge weight

ω

, where

ω

⁢

(

e

i

⁢

j

)

=

B

i

⁢

j

.
Clearly

G

is a

bipartite graph

, with

partitions

R

=

{

r

1

,

…

,

r

n

}

and

C

=

{

c

1

,

…

,

c

n

}

, since the only edges in

E

are between

r

i

and

c

j

for some

i

,

j

∈

{

1

,

…

,

n

}

. Furthermore, since

B

i

⁢

j

≥

0

, then

ω

⁢

(

e

)

>

0

for every

e

∈

E

.
For any

A

⊂

V

define

N

⁢

(

A

)

, the neighbourhood of

A

, to be the set of vertices

u

∈

V

such that there is some

v

∈

A

such that

(

u

,

v

)

∈

E

.
We claim that, for any

v

∈

V

,

∑

u

∈

N

⁢

(

{

v

}

)

ω

⁢

(

u

,

v

)

=

1

. Take any

v

∈

V

; either

v

∈

R

or

v

∈

C

. Since

G

is bipartite,

v

∈

R

implies

N

⁢

(

{

v

}

)

⊂

C

, and

v

∈

C

implies

N

⁢

(

{

v

}

)

⊂

R

. Now,

v

=

r

i

⟹

∑

u

∈

N

⁢

(

r

i

)

ω

⁢

(

r

i

,

u

)

=

∑

e

i

⁢

j

∈

E

j

=

1

,

n

ω

⁢

(

r

i

,

c

j

)

=

∑

B

i

⁢

j

≠

0

j

=

1

,

n

B

i

⁢

j

=

∑

j

=

1

n

B

i

⁢

j

=

1

v

=

c

j

⟹

∑

u

∈

N

⁢

(

c

j

)

ω

⁢

(

u

,

c

j

)

=

∑

e

i

⁢

j

∈

E

i

=

1

,

n

ω

⁢

(

r

i

,

c

j

)

=

∑

B

i

⁢

j

≠

0

i

=

1

,

n

B

i

⁢

j

=

∑

i

=

1

n

B

i

⁢

j

=

1

since

B

is doubly stochastic. Now, take any

A

⊂

R

. We have

∑

w

∈

N

⁢

(

A

)

v

∈

A

ω

⁢

(

v

,

w

)

=

∑

v

∈

A

∑

w

∈

N

⁢

(

{

v

}

)

ω

⁢

(

v

,

w

)

=

∑

v

∈

A

1

=

|

A

|

Let

B

=

N

⁢

(

A

)

. But then clearly

A

⊂

N

⁢

(

B

)

, by definition of neighbourhood. So

|

N

⁢

(

A

)

|

=

|

B

|

=

∑

w

∈

N

⁢

(

B

)

v

∈

B

ω

⁢

(

v

,

w

)

≥

∑

w

∈

A

v

∈

B

ω

⁢

(

v

,

w

)

=

∑

v

∈

N

⁢

(

A

)

w

∈

A

ω

⁢

(

v

,

w

)

=

|

A

|

So

|

N

⁢

(

A

)

|

≥

|

A

|

. We may therefore apply the graph-theoretic version of Hall’s

marriage theorem

to

G

to conclude that

G

has a

perfect matching

.
So let

M

⊂

E

be a perfect matching for

G

. Define an

n

×

n

matrix

P

by

P

i

⁢

j

=

{

1

if

⁢

e

i

⁢

j

∈

M

0

otherwise

Note that

B

i

⁢

j

=

0

implies

P

i

⁢

j

=

0

: if

B

i

⁢

j

=

0

, then

(

r

i

,

c

j

)

∉

E

, so

(

r

i

,

c

j

)

∉

M

, which implies

P

i

⁢

j

=

0

.
Further, we claim that

P

is a permutation matrix:

Let

i

be any row of

P

. Since

M

is a perfect matching of

G

, there exists

e

0

∈

M

such that

r

i

is an end of

e

0

. Let the other end be

c

j

for some

i

; then

P

i

⁢

j

=

1

.

Suppose

i

i

,

i

2

∈

{

1

,

…

,

n

}

with

i

1

≠

i

2

and

P

i

1

,

j

=

P

i

2

,

j

=

1

for some

j

. This implies

(

r

i

1

,

c

j

)

,

(

r

i

2

,

c

j

)

∈

M

, but this implies the vertex

i

is the end of two distinct edges, which contradicts the fact that

M

is a matching.

Hence, for each row and column of

P

, there is exactly one nonzero entry, whose value is 1. So

P

is a permutation matrix.
Define

λ

=

min

i

,

j

∈

{

1

,

…

,

n

}

⁡

{

B

i

⁢

j

∣

P

i

⁢

j

≠

0

}

. We see that

λ

>

0

since

B

i

⁢

j

≥

0

, and

P

i

⁢

j

≠

0

⟹

B

i

⁢

j

≠

0

. Further,

λ

=

B

p

⁢

q

for some

p

,

q

.
Let

D

=

B

-

λ

⁢

P

. If

D

=

0

, then

λ

=

1

and

B

is a permutation matrix, so we are done. Otherwise, note that

D

is nonnegative; this is clear since

λ

⁢

P

i

⁢

j

≤

λ

≤

B

i

⁢

j

for any

B

i

⁢

j

≠

0

.
Notice that

D

p

⁢

q

=

B

p

⁢

q

-

λ

⁢

P

p

⁢

q

=

λ

-

λ

⋅

1

=

0

.
Note that since every row and column of

B

and

P

sums to 1, that every row and column of

D

=

B

-

λ

⁢

P

sums to

1

-

λ

. Define

B

′

=

1

1

-

λ

⁢

D

. Then every row and column of

B

′

sums to 1, so

B

′

is doubly stochastic.
Rearranging, we have

B

=

λ

⁢

P

+

(

1

-

λ

)

⁢

B

′

. Clearly

B

i

⁢

j

=

0

implies that

P

i

⁢

j

=

0

which implies that

B

i

⁢

j

′

=

0

, so the zero entries of

B

′

are a superset of those of

B

′

. But notice that

B

p

⁢

q

′

=

1

1

-

λ

⁢

D

p

⁢

q

=

0

, so the zero entries of

B

′

are a strict superset of those of

B

.
We have decomposed

B

into a convex combination of a permutation matrix and another doubly stochastic matrix with strictly more zero entries than

B

. Thus we may apply this procedure repeatedly on the doubly stochastic matrix obtained from the previous step, and the number of zero entries will increase with each step. Since

B

has at most

n

2

nonzero entries, we will obtain a convex combination of permutation matrices in at most

n

2

steps.
Thus

B

is indeed expressible as a convex combination of permutation matrices.

References

G. Birkhoff, “Tres observaciones sobre el algebra lineal,” Univ. Nac. Tucumán Rev, Ser. A, no. 5, pp147–151. (1946)

Title

proof of Birkhoff-von Neumann theorem

Canonical name

ProofOfBirkhoffvonNeumannTheorem

Date of creation

2013-03-22 13:10:03

Last modified on

2013-03-22 13:10:03

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

14

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 15A51

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