Table Of ContentCOMBINATORIAL PROOFS OF LINEAR ALGEBRAIC IDENTITIES
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Mathematics
by Melanie Dennis
Guarini School of Graduate and Advanced Studies
DARTMOUTH COLLEGE
Hanover, New Hampshire
May 3, 2019
Examining Committee:
Peter Doyle, Chair
Peter Winkler
Rosa Orellana
Karen Yeats
F. Jon Kull, Ph.D.
Dean of the Guarini School of Graduate and Advanced Studies
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Abstract
Inthisthesis, weexaminetwodeterminantalidentities: theLewisCarrollIdentityand
the Dodgson/Muir Identity. The Lewis Carroll Identity expresses the determinant of
a matrix in terms of subdeterminants obtained by deleting one row and column or
a pair of rows and columns. Using the Matrix Tree Theorem, we convert this into
an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll
Identity, the Forest Identity involves no minus signs. Using the Involution Principle,
we can pull back Zeilberger’s proof of the Lewis Carroll Identity to a bijective proof of
the Forest Identity. This bijection is implemented by the Red Hot Potato algorithm,
so called because the way edges get tossed back and forth between the two forests is
reminiscent of the children’s game of hot potato. We examine in detail the connection
between the Red Hot Potato algorithm and Zeilberger’s proof. The Dodgson/Muir
Identity is a generalization of the Lewis Carroll Identity that expresses the determi-
nant of a matrix in terms of subdeterminants obtained by deleting sets of k−1 rows
and columns or sets of k rows and columns. Again, we find the equivalent identity
involving sums over k-tuples of forests. We give a direct proof of the Dodgson/Muir
Identity using pairwise iterations of the Red Hot Potato algorithm.
ii
Acknowledgments
First and foremost, I would like to thank my amazing advisor, Peter Doyle, for all of
his support during my time here at Dartmouth. He is absolutely brilliant, helped me
work through the ideas presented here in this thesis, and came up with all of the best
bijection names. Without him, I most certainly would not have made it through this
program. He is a huge proponent of work-life balance and encouraged me to invest
time in my life outside of math, so that I was much happier in the program, and also
hugely more productive in my work. Thank you Peter!
I would also like to thank my committee members for taking the time to read
through the thesis and give me valuable feedback. Because of them, this thesis is a
lot more readable.
Finally, I would like to thank the amazing support system of friends and family
that got me through these five years. David, Kate, Angie, Sara, and Doug, just to
name a few, you all were there for me when I did not believe in myself, and I would
not have finished without you. And last but certainly not least, Tim, my wonderful
fianc´e and biggest cheerleader, thank you for listening to all of my ideas even when
you didn’t understand them, and for encouraging me when I wanted to give up. This
thesis is dedicated to you.
iii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction 1
1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Lewis Carroll Identity . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Dodgson/Muir Identity . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The Matrix Tree Theorem . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Involution Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Lewis Carroll and the Red Hot Potato 16
2.1 Connecting Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Red Hot Potato algorithm . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Proof of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1
2.5 Identity Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
iv
3 Connection to Zeilberger 41
3.1 Zeilberger’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Alternate Definition of φ . . . . . . . . . . . . . . . . . . . . . . . . 48
1
4 Dodgson/Muir Identity 59
4.1 Connecting Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Generalized Red Hot Potato algorithm . . . . . . . . . . . . . . . . . 69
4.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.3 Generalized Red Hot Potato algorithm . . . . . . . . . . . . . 74
4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Proof that φ is a sign-reversing involution . . . . . . . . . . . . . . . 78
1
4.5 Proofs of Forest and Lewis Carroll Identities . . . . . . . . . . . . . . 84
5 Future Work 87
5.1 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Other Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References 91
v
Chapter 1
Introduction
This dissertation focuses on providing graph theoretic bijections to linear algebraic
determinantal identities. We will begin by reviewing some definitions.
Section 1.1
Basic Definitions
Definition 1.1. A directed graph is a set of nodes V and set of ordered pairs
(s,t) ∈ V × V called arcs. An arc can be visualized as an arrow that starts at s
and terminates at t. Because we will be dealing primarily with directed graphs (as
opposed to undirected graphs) in this thesis, we will generally call arcs edges.
Definition 1.2. A tree is a connected directed graph with no directed cycles such
that no node has more than one edge coming out of it. A root is a node in a tree
such that there exists a directed path from all other nodes to the root. Note that a
root cannot have any out-edges, or else it would form a directed cycle. A forest is a
disjoint union of trees.
1
1.2 History
Notice that, in using these definitions, a forest is a directed graph with no cycles
in which every node has either one out-edge or no out-edges. The roots in the forest
are exactly those nodes with no out-edges.
Example 1.3.
1 1
0 2 0 2
4 3 4 3
The forest on the left is a two-forest (a forest with two trees). One tree is rooted
at node 4 and one is rooted at node 0. The forest on the right is a three-forest. One
tree is rooted at 0, one at 1, and one at 2.
Definition 1.4. A path from node i to node j is called a meta-edge i → j.
Example 1.5. In the following example, the edges 1 → 4, 4 → 3, 3 → 2 together form
the meta-edge 1 → 2, and the edge 2 → 1 forms the meta-edge 2 → 1.
1 1
0 2 0 2
4 3 4 3
Section 1.2
History
In this thesis, we will be focusing on proving two determinantal identities: the Lewis
Carroll Identity and the Dodgson/Muir Identity. Before we state the identities, we
need two further definitions.
2
1.2 History
Definition 1.6. Let U and W be sets of integers of the same size, and let M be a
matrix. Then M is the submatrix of M with the rows corresponding to U removed
U,W
and the columns corresponding to W removed.
Definition 1.7. Let U and W be sets of integers of the same size, and let M be a
matrix. Then M[U,W] is the submatrix of M containing only the rows corresponding
to the elements in U and the columns corresponding to the elements in W (with the
rest of the rows and columns in M removed).
1.2.1. Lewis Carroll Identity
Also known as the Desnanot-Jacobi Identity or Dodgson’s determinant evaluation
rule, the Lewis Carroll Identity relates the determinant of a matrix to that of its
minors.
Theorem 1.8. Lewis Carroll Identity. Let M be a square matrix. Then
det(M)·det(M ) = det(M )·det(M )−det(M )·det(M ).
{1,2},{1,2} {2},{2} {1},{1} {2},{1} {1},{2}
This is for any n × n matrix. Lagrange first proved the identity for n = 3 in
1773 ([11], pg 39). In 1819, Desnanot proved it for n ≤ 6, and Minding also proved
the same for small n in 1829 ([11], pg 142, 197). Finally, in 1833, Jacobi proved the
identity for any n ([11], pg 208-209). All of these proofs were algebraic and used the
transposeofthecofactormatrix. Aproofcanbefoundonpages112-113ofBressoud’s
book, [4].
ThoughmostproofsoftheLewisCarrollIdentityarealgebraic, in1997, Zeilberger
created a combinatorial proof [14]. That proof will be given in Chapter 3.
3
