ABSTRACT TAYLOR, VALERIE EOWYN. The Birkhoff-von Neumann Decomposition and its Applications. (Under the direction of Dr. Arvind Krishna Saibaba). This paper explores the Birkhoff-von Neumann decomposition theorem which is a celebrated theorem applicable to a specific class of matrices, called doubly stochastic matrices. The Birkhoff- von Neumann decomposition has many application ranging from theoretical areas such as matrix approximation to applied areas such as graph isomorphisms and assignment. The purpose of this paper is to review the literature, to collect and organize, various statements and applications of this theorem. This paper will start with presenting two different and equivalent statements of the theorem. We will present proofs for both of these statements of the theorem and highlight the connections between them. We then go into the applications of this theorem. We mention several theoretical applications where the Birkhoff-von Neumann is used in the proof of other results. We then address some more applied results such as graph isomorphism and the assignment problem. The Birkhoff-von Neumann Decomposition and its Applications by Valerie Eowyn Taylor A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Master of Science Mathematics Raleigh, North Carolina 2018 APPROVED BY: Dr. Arvind Krishna Saibaba Dr. Ernie Stitzinger Chair of Advisory Committee Dr. Agnes Szanto ii BIOGRAPHY The author was born in Chapel Hill, NC on April 8th, 1991. They were homeschooled by their mother until attending public high school in 9th grade at Cedar Ridge High School in Hillsborough, NC. They then completed their undergraduate career at the University of North Carolina at Wilm- ington, receiving their Bachelor of Arts in Mathematics Education. Upon graduating, they began teaching math at John T. Hoggard High School in Wilmington. They taught at JTH for three years until entering graduate school at North Carolina State University. They plan on returning to the education field upon completion of the graduate program at NCSU. iii TABLE OF CONTENTS 1. Introduction ......................................................................... 1 2. Birkhoff-von Neumann Decomposition....................................... 2 3. Applications ......................................................................... 8 3.1. Von Neumann Trace Inequality....................................................... 8 3.2. Hoffman-Wielandt Theorem........................................................... 13 3.3. Other Applications....................................................................... 15 Graph Isomorphisms ........................................................... 15 Assignment Problem ........................................................... 16 Majorization ...................................................................... 18 Fan Dominance Principle ..................................................... 19 4. Conclusion............................................................................ 20 References ............................................................................ 21 APPENDICES ...................................................................... 22 Appendix A: Additional Proofs....................................................... 23 Appendix B: Examples ................................................................. 24 1 1. Introduction Doubly stochastic matrices are square matrices with non-negative entries such that all the rows and columns sum to 1. Doubly stochastic matrices have many applications. These matrices are used in intercity population migration models. An example of this would be that there are cities C1;:::;Cn with n ≥ 2 and each day a constant fraction aij of the current population of city j moves to city i for all distinct i; j 2 f1; : : : ; ng. Problems like this are necessary for issues such as planning city services and capital investment. These migration models quickly become complication and doubly stochastic matrices can be extremely useful in the calculations [4]. Doubly stochastic matrices are also used in Markov Chains and modeling problems in economics and operations research. Another application related to modeling deals with communication theory and satellites orbiting the earth [2]. This paper will explore Birkhoff-von Neumann decomposition and various applications of the theorem. We will first present two different, and equivalent, statements of the theorem along with two different proofs. Then we will look at how this result impacted the development of other results also well as some applied applications of the theorem. Throughout this document let A = [aij] be an n × n matrix in R, unless otherwise stated. We denote by π a permutation of the integers f1; : : : ; ng, and it's entries by π(i) for i = 1; : : : ; n. Denote the columns of the n × n identity matrix by e1; : : : ; en. Associated with every permutation π is the permutation matrix P given by P = : eπ(1) : : : eπ(n) 2 2. Birkhoff-von Neumann Decomposition The Birkhoff-von Neumann Decomposition theorem is an important result for a special class of matrices known as doubly stochastic, a definition of which follows below. Definition 1 (Doubly Stochastic Matrices). A doubly stochastic matrix is a square n × n matrix A, with non-negative entries aij ≥ 0 for i; j = 1; : : : ; n and n n X X aij = 1; j = 1; : : : ; n and aij = 1; i = 1; : : : n: i=1 j=1 This definition simply means that all the columns and the rows sum to 1. Alternatively, given T n a doubly stochastic matrix A and the vector of all ones e = (1;:::; 1) 2 R , then Ae = e and eT A = eT . This second definition implies that 1 is always an eigenvalue value of a doubly stochastic matrix, and the corresponding right and left eigenvector is e. Given this definition, we can state the first version of the Birkhoff-von Neumann decomposition theorem. Theorem 2 (Birkhoff-von Neumann Decomposition). Let A be a doubly stochastic ma- Pk trix, there exist constants α1; α2; : : : ; αk 2 (0; 1) with i=1 αi = 1 and permutation matrices P1;P2;:::;Pk such that A = α1Pi + ::: + αkPk: That is, a doubly stochastic matrix can be expressed as a convex combination of permutation matrices. Conversely, a single permutation matrix is also a doubly stochastic matrix. Note the summation is what is referred to as the Birkhoff-von Neumann decomposition. Before we can prove this theorem, we need to establish the following lemma. 3 Lemma 3. Let A be a n × n doubly stochastic matrix that is not the identity matrix. There is a permutation π of f1; : : : ng that is not the identity permutation and is such that a1π(1) ··· anπ(n) > 0: This means that we can find n nonzero elements of A, one in each column. Recall that all of the entries in A are positive, and this lemma is saying that their product is positive, which is simply implying that none of the entries are zero. The proof of this lemma is located in Appendix A, so that we can directly give the proof of the Birkhoff-von Neumann theorem. The following proof is adapted from the proof presented by Marshall, Olkin and Arnold [6]. Proof of Theorem 2. Let A be doubly stochastic. If A is a permutation matrix, there is nothing to prove. So, assume that A is not a permutation matrix. Let π be a permutation of (1; : : : ; n) that is not the identity permutation, such that the product a1π(1)a2π(2) : : : anπ(n) 6= 0, whose existence is ensured by Lemma 3. Denote the corresponding permutation matrix by P1. Let c1 = minfa1π(1); : : : ; anπ(n)g and define R by A = c1P1 + R. Note that c1 ≤ 1 since A is doubly stochastic. Also note that c1 6= 0 since none of the values fa1π(1); : : : ; anπ(n)g can equal 0 since their product is not 0. Because c1P1 has element c1 in positions 1π(1); 2π(2); : : : ; nπ(n) and A has elements a1π(1); : : : anπ(n) in the corresponding positions, the choice of c1 = minfa1π(1); : : : ; anπ(n)g ensures that alπ(l) −c1 ≥ 0, with equality for some l. Consequently, R has non-negative elements, since rlπ(l) = alπ(l) − c1, and contains at least one more zero element than A, since for some l we know that alπ(l) = c1 which implies rlπ(l) = 0. Observe that for e = (1; 1;:::; 1)T we have that e = Ae = (c1P1 + R)e = c1P1e + Re = c1e + Re: 4 Now we have two cases to look at. First consider if c1 = 1. This implies that R = 0 and A = P1 so A is already a permutation matrix and the desired decomposition is trivial. For the second case consider c1 < 1. From our earlier statement that e = c1e + Re we can say that e = A e where A = R . A is doubly stochastic since all entries are positive and A e = e, 1 1 1−c1 1 1 which is the definition of doubly stochastic. In this case, we apply the same procedure to A1 to continue the decomposition. Each time we reduce the number of nonzero entries in the remainder, until we get the zero matrix. Note that each time we pick a permutation of f1; : : : ; ng we necessarily pick a permutation that is not the identity. If the identity permutation is the only one available, then A = I as shown in the proof of Lemma 3, and hence the decomposition is done. Consequently, for some k, when the remainder is 0, we have A = c1P1 + ::: + ckPk where each Pi is a permutation matrix. In remains to observe that e = Ae = c1P1e + ::: + ckPke = (c1 + : : : ck)e Pk which implies that i=1 ci = 1. This completes the proof. It is relevant here to note a bound on the number of iterations needed to reach the decomposition and a bound on k, the number of summands. We know that a doubly stochastic matrix has at most n2 − n zero entries. This is because there are a total of n2 entries and there must be at least one nonzero entry in each column, meaning a least n non-zero entries. Since there are at most n2 − n zero entries this implies there are at most n2 − n iterations in the decomposition process. After this many iterations we would have n2 − n + 1 summands, that is k ≤ n2 − n + 1. This bound has since been improved upon by several sources. For reference on these sources consult [1, Section II pg. 38] and [6, Section 2 Theorem F.2].