Is the Birkhoff von Neumann Theorem true for infinite matrices

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The Birkhoff von Neumann Theorem states that any $ntimes n$ doubly stochastic matrix is a convex sum of permutation matrices.

Is this true for $mathbbN times mathbbN$ matrices as well? (if yes, then can you give a reference?)

linear-algebra discrete-mathematics game-theory

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asked Jul 31 at 9:54

Yoav Rodeh

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The Birkhoff von Neumann Theorem states that any $ntimes n$ doubly stochastic matrix is a convex sum of permutation matrices.

Is this true for $mathbbN times mathbbN$ matrices as well? (if yes, then can you give a reference?)

linear-algebra discrete-mathematics game-theory

share|cite|improve this question

asked Jul 31 at 9:54

Yoav Rodeh

111

add a comment | 

up vote
1
down vote

favorite

1

up vote
1
down vote

favorite

1

1

The Birkhoff von Neumann Theorem states that any $ntimes n$ doubly stochastic matrix is a convex sum of permutation matrices.

Is this true for $mathbbN times mathbbN$ matrices as well? (if yes, then can you give a reference?)

linear-algebra discrete-mathematics game-theory

share|cite|improve this question

asked Jul 31 at 9:54

Yoav Rodeh

111

The Birkhoff von Neumann Theorem states that any $ntimes n$ doubly stochastic matrix is a convex sum of permutation matrices.

Is this true for $mathbbN times mathbbN$ matrices as well? (if yes, then can you give a reference?)

linear-algebra discrete-mathematics game-theory

share|cite|improve this question

asked Jul 31 at 9:54

Yoav Rodeh

111

share|cite|improve this question

share|cite|improve this question

asked Jul 31 at 9:54

Yoav Rodeh

111

asked Jul 31 at 9:54

Yoav Rodeh

111

asked Jul 31 at 9:54

Yoav Rodeh

111

111

add a comment | 

add a comment | 

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I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,

I read:

Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).

The reference Mirsky (1) is to the following paper:

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.

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edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

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1 Answer
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active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

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active

oldest

votes

up vote
2
down vote

I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,

I read:

Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).

The reference Mirsky (1) is to the following paper:

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.

share|cite|improve this answer

edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

45.9k348110

add a comment | 

up vote
2
down vote

I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,

I read:

Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).

The reference Mirsky (1) is to the following paper:

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.

share|cite|improve this answer

edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

45.9k348110

add a comment | 

up vote
2
down vote

up vote
2
down vote

I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,

I read:

Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).

The reference Mirsky (1) is to the following paper:

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.

share|cite|improve this answer

edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

45.9k348110

I don't know the exact answer to your question but I found a reference that should be useful. In L. Mirsky's book Transversal Theory (Academic Press, 1971, ISBN 0-12-498550-5), on p.213,

I read:

Birkhoff's theorem has been extended to infinite d.s. matrices, with the notion of convex closure in a suitable topological vector space replacing that of convex hull. For references to work in this field, see Mirsky (1).

The reference Mirsky (1) is to the following paper:

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices,
Z. Wahrscheinlichkeitstheorie 1 (1962–3), 319–334.

share|cite|improve this answer

edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

45.9k348110

share|cite|improve this answer

share|cite|improve this answer

edited Jul 31 at 13:33

edited Jul 31 at 13:33

edited Jul 31 at 13:33

answered Jul 31 at 12:10

bof

45.9k348110

answered Jul 31 at 12:10

bof

45.9k348110

answered Jul 31 at 12:10

bof

45.9k348110

45.9k348110

add a comment | 

add a comment | 

 

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