The MatrixTree Theorem


 Gilbert Hill
 1 years ago
 Views:
Transcription
1 The MatrixTree Theorem Christopher Eur March 22, 2015 Abstract: We give a brief introduction to graph theory in light of linear algebra. Our results culminates in the proof of MatrixTree Theorem. 1 Preliminaries We define preliminary definitions and give a brief list of facts from linear algebra without proof. Definition 1.1. For k Z 0, denote by [k] the set {1, 2,..., k} ([0] := ). Definition 1.2. Let X be a set, then denote by ( ) X k the set of kelement subsets of X. Denote by ( ( ) X k ) the set of kelement multisets (sets with repeated elements) of X. Example 1.3. Let X = {a, b, c, d}. Then {a, c}, {b, c} ( ) ( X 2, {a, a} / X ) 2, and {a, b, d}, {a, a, b}, {c, c, c} ( ( ) X 3 ) Proposition 1.4. A, a n n matrix, that is real symmetric has a orthonormal eigenbasis (u 1, u 2,..., u n ), u i R n where u t i u j = δ ij (Kronecker delta). Moreover, if U = [u 1... u n ] is a n n matrix whose columns are u 1,..., u n, then U 1 = U t and U t AU is diagonal. Definition 1.5. Let A be a m n matrix with m n, and S ( [n] m). Denote by A[S] the m m submatrix of A formed by taking columns of A indexed by S. Likewise, if B is a n m matrix with m n, then denote by B[S] the m m submatrix of B formed by taking rows of B indexed by S. Observation 1.6. Note that (A[S]) t = A t [S] 1
2 Example 1.7. If Then A = [ ] , B = A[S] = [ ] 2 7, B[S] = 4 0 ( [4], S = {2, 3} 2 [ 2 ] Theorem 1.8. (CauchyBinet Formula) Let A, B be m n, n m matrices, respectively, and m n. Then det(ab) = det(a[s]) det(b[s]) S ( [n] m) Proposition 1.9. Let A be an n n matrix such that the sum of entries in each row and column is zero, and A 0 be the matrix obtained by removing the last row and column of A. Then the coefficient of x in det(a xi) is equal to n det(a 0 ). ) 2 Walks on Graphs We present elementary graph theory on walks on graphs. Definition 2.1. A (finite) graph G = (V (G), E(G)) consists of a vertex set V (G) = {v 1,..., v p } and an edge set E(G) = {e 1,..., e q } with a function ϕ : E ( ( V 2) ). Moreover, for e E(G) and v, v V (G), if ϕ(e) = {v, v } then e connects or is incident to v and v ; if e that connects v, v, then v and v are adjacent; if ϕ(e) = {v, v} then e is called a loop Example 2.2. v2 e1 e2 e3 v3 v1 is a graph G = (V = {v1, v2, v3}, E = {e1, e2, e3}) where ϕ(e1) = ϕ(e2) = {v2, v3}, ϕ(e3) = {v1, v2} 2
3 NOTE: For the remainder of the article we will say our graph G has p many vertices and q many edges unless stated otherwise. Definition 2.3. The adjacency matrix of graph G, denoted A(G) (or just A if G is clear), is the p p matrix where A ij =(number of edges incident to v i and v j ) Example 2.4. In the Example 2.2, we have A = Observation 2.5. Let G be a graph. Then we can easily see that A(G) is (real) symmetric, and the trace is the number of loops in G. Definition 2.6. A walk of length l on a graph G from vertex u to vertex v is a sequence v 1, e 2, v 2, e 2,..., v l, e l, v l+1 such that v i s are in V (G), v 1 = u, v l+1 = v, and e i s are in E(G) such that e i connects v i and v i+1. Now we have our first elementary result: Theorem 2.7. Let G be a graph, A = A(G). Then (A l ) ij =(number of walks from v i to v j for all l N) Proof) We induct on l. l = 1 case is trivial. (A l ) ij = p k=1 A ik(a l 1 ) kj and A ik is the number of walks of length 1 from v i to v k and by induction (A l 1 ) kj is the number of walks of length l 1 from v k to v i, so summing over k we have our desired result. We also know that A(G) is real symmetric, so let λ 1,..., λ p be the eigenvalues and u 1,..., u p be the orthonormal basis that corresponds to them respectively, and let U be the orthogonal matrix formed by u i s as the columns. Then we have the following refinement of the previous theorem in more algebraic terms: Corollary 2.8. Let u ij be (i, j)th coordinate of U. Then (A l ) ij = p u ik u jk λ l k k=1 and defining closed walk as a walk that starts and ends at the same vertex, we have (number of closed walks of length l) = Tr(A l ) = λ l λ l p 3
4 Proof) Let D = Diag(λ 1,..., λ p ). Then both statements follow immediately from U D l U t = A l (noting that Tr(AB) = Tr(BA)). We give an application of this corollary to completely graph as an example: Definition 2.9. A complete graph K p is a graph G with p vertices such that there exists exactly one edge connecting v i to v j for all 1 i < j p. Example Below are K 3, K 4, and K 5 : Proposition Let K p be a complete graph, A = A(K p ). Then (A l ) ii = 1 p ((p 1) l + (p 1)( 1) l) Proof) By symmetry, note that (A l ) ii s are equal for all i = 1,..., p. So, we show Tr(A l ) = (p 1) l + (p 1)( 1) l. Let J be a p p matrix with all entries equal to 1. It is easy to check that J has eigenvalues 0 (with multiplicity p 1) and p (with multiplicity 1), so J I has eigenvalues 1 (with multiplicity p 1) and p 1 (with multiplicity 1). And since A(K p ) is exactly J I, applying our previous corollary we have our desired result. 3 The MatrixTree Theorem For this section, we assume that G has no loops because they are completely irrelevant to our discussion. We also assume G is connected, that is, there exists a walk between any two vertices of G. 4
5 Definition 3.1. A cycle is a closed walk with no repeated vertices or edges except for the first and last vertex. A tree is a (connected) graph that has no cycles. Observation 3.2. It s useful to observe that the following are equivalent: (i) G is a tree, (ii) G is connected and has p 1 edges, (iii) G has no cycles and has p 1 edges. Definition 3.3. Let G be a graph. A subgraph G of G is a graph with V (G ) V (G), E(G ) E(G). A subgraph G is a spanning subgraph if V (G ) = V (G). A subgraph G is a spanning tree if it is a spanning subgraph that is a tree. Definition 3.4. Let G be a graph. The complexity of G, denoted κ(g), is the number of spanning trees of G. Example 3.5. Going back to K 3, K 4, and K 5 : It is not hard to count combinatorially and confirm that κ(k 3 ) = 3, κ(k 4 ) = 16, κ(k 5 ) = 125. For example, for K 5, the three types of trees with 5 vertices are: v1 v2 v3 v4 v5 v1 v2 v3 v5 v4 v1 v5 v3 v2 v4 5
6 And there are 5!/2 many spanning trees of K 5 of the first type, ( 5 2) (3)(2) many of the second type, and 5 of the last type. Summing up we have 125. From this we may guess that the formula for κ(k p ) = p p 2. The remainder of our discussion in this section will culminate in proving this result. Recall our notation that a generic graph G has vertices {v 1,..., v p } and edges {e 1,..., e q }. Definition 3.6. Let G be a graph. We can give G an orientation. That is, for ϕ(e) = {u, v}, choose one of two ordered pairs (u, v), (v, u). Say we chose (u, v), then we call u the initial vertex and v the final vertex of e, and e is directed from u to v. (Intuitively, this is putting an arrow on e pointing from u to v). We first define two more algebraic tools (matrices) that will aid us: Definition 3.7. Let G be a graph with an orientation. The incidence matrix M(G) = M is the p q matrix defined by: 1 if the edge e j has initial vertex v i M ij = 1 if the edge e j has final vertex v i 0 otherwise Definition 3.8. Let G be a graph with an orientation and A = A(G). The Laplacian matrix L(G) = L is the p p matrix defined by: { Aij if i j M ij = deg(v i ) if i = j Example 3.9. Below is an oriented graph v2 e1 e2 v3 e3 e5 v1 e4 v4 6
7 with: M = , L = Observation In M, each column entries sum to zero, so rows of M sum to zero vector, hence rank(m) < p. In L, entries in each column or row sum to zero, and L is symmetric. Lastly, if G is regular of degree d, that is deg(v i ) = d for all v i V (G), then L = di A(G). We can now state our main theorem: Theorem (MatrixTree Theorem) Let G be (finite connected) graph (without loops), and let L = L(G). Denote by L 0 the matrix obtained by removing the last row and column of L. Then Our proof requires two lemmas: det(l 0 ) = κ(g) Lemma Let G be a graph, give G any orientation, and let L = L(G), M = (G). Then L = MM t. Proof) We have M ij = q M ik M jk k=1 If i j, for each k, if e k connects v i and v j exactly one of M ik, M jk is 1, and the other 1. If i = j, either M ik, M jk are both zero (if e k is not incident to v i ) and both 1 or 1 if e k is incident to v i. Definition Let G be a graph, M = M(G), and M 0 := (p 1) q matrix obtained by removing the last row of M. For S ( [q] p 1), we define S(E) to be a subset of edges indexed by S (i.e. S(E) = {e i } i S {e 1,..., e q }), and define p 1 p 1 matrix M 0 [S] as in definition 1.5. Moreover, by graph formed by S(E) (which we also denote as S(E)), we mean graph with edges in S(E) and vertices incident to an edge in S(E). Lemma If S(E) forms a spanning tree, then det M 0 [S] = ±1, if S(E) does not, then det M 0 [S] = 0. 7
8 Proof) If S(E) does not form a spanning tree, since we have p 1 edges there must be a subset R of S such that R(E) forms a cycle, so let this cycle be f 1,..., f j (a walk without the vertices written down). Then multiplying 1 if necessary, the columns of M corresponding to f i s sum to zero, hence det M 0 [S] = 0. Now suppose S(E) forms a spanning tree. Let e be an edge of S(E) incident to v p, then the column of M 0 [S] corresponding to e contains exactly one nonzero entry which is ±1. So, If M 0 is the p 2 p 2 matrix obtained by removing the two and column containing the nonzero entry of column e, by determinant expansion det M 0 [S] = ± det M 0. Now consider G a graph (in fact a tree) obtained by contracting the edge e to a single vertex (so v p is now identified with the vertex e connected v p to.) M(G ) p 1 p 2 matrix formed by removing the column e, and so M 0 is the matrix formed by removing the bottom row of M(G ). And thus by induction det M 0 = ±1, and this completes our proof. Proof of the MatrixTree Theorem Since L = MM, we have L 0 = M 0 M0 t. By CauchyBinet formula, we have det L 0 = (det M 0 [S])(det M0[S]) t = (det(m 0 [S])) 2 ( (A[S]) t = A t [S]) S S ( [q] p 1) and (det(m 0 [S])) 2 is 1 if S(E) forms a spanning tree and 0 otherwise, and a tree with p vertices has p 1 edges, so summing over S ( [q] p 1), RHS exactly counts the number of spanning trees of G. Corollary Let G be a (finite connected) graph (without loops) with p vertices, and let λ 1,..., λ p be the eigenvalues of L(G) with λ p = 0, then κ(g) = 1 p λ 1λ 2 λ p 1 Proof) First we note that λ p = 0 is possible since each rows (and columns) sum to zero in L(G) ( Observation 3.10). And det(l xi) = (λ 1 x)(λ 2 x) (λ p 1 x)( x). So by Proposition 1.9 we have λ 1... λ p 1 ( coeff. of x) equals p det(l 0 ), and the corollary thus follows. 8
9 Corollary Let G be a regular graph of degree d, and let the eigenvalues of A(G) be λ 1,..., λ p with λ p = d. Then, κ(g) = 1 p (d λ 1)(d λ 2 ) (d λ p 1 ) Proof) Follows immediately from L = di A(G) and the previous corollary. Now, as we promised, we have: Theorem κ(k p ) = p p 2 Proof) K p is regular of degree d, and A(K p ) has eigenvalues 1 with multiplicity p 1 and p 1 with multiplicity 1. So from our Corollary 3.16 we have κ(k p ) = 1 p ((p 1) ( 1))p 1 = p p 2, as desired. 4 References Stanley, Richard P. Topics in Algebraic Combinatorics. Version 1 Feb rstan/algcomb/algcomb.pdf 9
ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact.
ORIE 6334 Spectral Graph Theory September 8, 2016 Lecture 6 Lecturer: David P. Williamson Scribe: Faisal Alkaabneh 1 The MatrixTree Theorem In this lecture, we continue to see the usefulness of the graph
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 19
832: Algebraic Combinatorics Lionel Levine Lecture date: April 2, 20 Lecture 9 Notes by: David Witmer MatrixTree Theorem Undirected Graphs Let G = (V, E) be a connected, undirected graph with n vertices,
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More informationSmith Normal Form and Combinatorics
Smith Normal Form and Combinatorics p. 1 Smith Normal Form and Combinatorics Richard P. Stanley Smith Normal Form and Combinatorics p. 2 Smith normal form A: n n matrix over commutative ring R (with 1)
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationSpectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics
Spectral Graph Theory and You: and Centrality Metrics Jonathan Gootenberg March 11, 2013 1 / 19 Outline of Topics 1 Motivation Basics of Spectral Graph Theory Understanding the characteristic polynomial
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More information7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4
7 Matrix Operations Copyright 017, Gregory G. Smith 9 October 017 The product of two matrices is a sophisticated operations with a wide range of applications. In this chapter, we defined this binary operation,
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 16, 2007 Abstract This paper gives a definition of the determinant and lists many of its wellknown properties Volumes of parallelepipeds are
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationMATH 1210 Assignment 4 Solutions 16RT1
MATH 1210 Assignment 4 Solutions 16RT1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,
More informationLinear Algebra II. Ulrike Tillmann. January 4, 2018
Linear Algebra II Ulrike Tillmann January 4, 208 This course is a continuation of Linear Algebra I and will foreshadow much of what will be discussed in more detail in the Linear Algebra course in Part
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
More informationLecture 23: Trace and determinants! (1) (Final lecture)
Lecture 23: Trace and determinants! (1) (Final lecture) Travis Schedler Thurs, Dec 9, 2010 (version: Monday, Dec 13, 3:52 PM) Goals (2) Recall χ T (x) = (x λ 1 ) (x λ n ) = x n tr(t )x n 1 + +( 1) n det(t
More informationdet(ka) = k n det A.
Properties of determinants Theorem. If A is n n, then for any k, det(ka) = k n det A. Multiplying one row of A by k multiplies the determinant by k. But ka has every row multiplied by k, so the determinant
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationOn the adjacency matrix of a block graph
On the adjacency matrix of a block graph R. B. Bapat StatMath Unit Indian Statistical Institute, Delhi 7SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More information1.10 Matrix Representation of Graphs
42 Basic Concepts of Graphs 1.10 Matrix Representation of Graphs Definitions: In this section, we introduce two kinds of matrix representations of a graph, that is, the adjacency matrix and incidence matrix
More informationMath 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.
Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses
More informationGraph fundamentals. Matrices associated with a graph
Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n
More informationRefined Inertia of Matrix Patterns
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 24 2017 Refined Inertia of Matrix Patterns Kevin N. Vander Meulen Redeemer University College, kvanderm@redeemer.ca Jonathan Earl
More informationA NEW INTERPRETATION OF THE MATRIX TREE THEOREM USING WEAK WALK CONTRIBUTORS AND CIRCLE ACTIVATION HONORS THESIS
A NEW INTERPRETATION OF THE MATRIX TREE THEOREM USING WEAK WALK CONTRIBUTORS AND CIRCLE ACTIVATION HONORS THESIS Presented to the Honors College of Texas State University in Partial Fulfillment of the
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationSpectral radius, symmetric and positive matrices
Spectral radius, symmetric and positive matrices Zdeněk Dvořák April 28, 2016 1 Spectral radius Definition 1. The spectral radius of a square matrix A is ρ(a) = max{ λ : λ is an eigenvalue of A}. For an
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
More informationEnergy of Graphs. Sivaram K. Narayan Central Michigan University. Presented at CMU on October 10, 2015
Energy of Graphs Sivaram K. Narayan Central Michigan University Presented at CMU on October 10, 2015 1 / 32 Graphs We will consider simple graphs (no loops, no multiple edges). Let V = {v 1, v 2,..., v
More informationMathematical Methods wk 2: Linear Operators
John Magorrian, magog@thphysoxacuk These are workinprogress notes for the secondyear course on mathematical methods The most uptodate version is available from http://wwwthphysphysicsoxacuk/people/johnmagorrian/mm
More informationFORBIDDEN MINORS FOR THE CLASS OF GRAPHS G WITH ξ(g) 2. July 25, 2006
FORBIDDEN MINORS FOR THE CLASS OF GRAPHS G WITH ξ(g) 2 LESLIE HOGBEN AND HEIN VAN DER HOLST July 25, 2006 Abstract. For a given simple graph G, S(G) is defined to be the set of real symmetric matrices
More informationAn Algorithmist s Toolkit September 10, Lecture 1
18.409 An Algorithmist s Toolkit September 10, 2009 Lecture 1 Lecturer: Jonathan Kelner Scribe: Jesse Geneson (2009) 1 Overview The class s goals, requirements, and policies were introduced, and topics
More informationk=1 ( 1)k+j M kj detm kj. detm = ad bc. = 1 ( ) 2 ( )+3 ( ) = = 0
4 Determinants The determinant of a square matrix is a scalar (i.e. an element of the field from which the matrix entries are drawn which can be associated to it, and which contains a surprisingly large
More informationMath 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank
Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank David Glickenstein November 3, 4 Representing graphs as matrices It will sometimes be useful to represent graphs
More informationDETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER R.H. HANUSA AND THOMAS ZASLAVSKY Abstract. We investigate the least common multiple of all subdeterminants,
More informationAlgorithms to Compute Bases and the Rank of a Matrix
Algorithms to Compute Bases and the Rank of a Matrix Subspaces associated to a matrix Suppose that A is an m n matrix The row space of A is the subspace of R n spanned by the rows of A The column space
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationDeterminants. Samy Tindel. Purdue University. Differential equations and linear algebra  MA 262
Determinants Samy Tindel Purdue University Differential equations and linear algebra  MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationLaplacian and Random Walks on Graphs
Laplacian and Random Walks on Graphs Linyuan Lu University of South Carolina Selected Topics on Spectral Graph Theory (II) Nankai University, Tianjin, May 22, 2014 Five talks Selected Topics on Spectral
More informationChapter 2 Spectra of Finite Graphs
Chapter 2 Spectra of Finite Graphs 2.1 Characteristic Polynomials Let G = (V, E) be a finite graph on n = V vertices. Numbering the vertices, we write down its adjacency matrix in an explicit form of n
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationBoolean InnerProduct Spaces and Boolean Matrices
Boolean InnerProduct Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationAn Algorithmist s Toolkit September 15, Lecture 2
18.409 An Algorithmist s Toolkit September 15, 007 Lecture Lecturer: Jonathan Kelner Scribe: Mergen Nachin 009 1 Administrative Details Signup online for scribing. Review of Lecture 1 All of the following
More informationThe Structure of the Jacobian Group of a Graph. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College
The Structure of the Jacobian Group of a Graph A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the
More informationNetworks and Matrices
BowenLanford Zeta function, Math table talk Oliver Knill, 10/18/2016 Networks and Matrices Assume you have three friends who do not know each other. This defines a network containing 4 nodes in which
More informationIdentities for minors of the Laplacian, resistance and distance matrices
Identities for minors of the Laplacian, resistance and distance matrices R. B. Bapat 1 Indian Statistical Institute New Delhi, 110016, India email: rbb@isid.ac.in Sivaramakrishnan Sivasubramanian Department
More informationSPRING OF 2008 D. DETERMINANTS
18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants
More informationDeterminants. Beifang Chen
Determinants Beifang Chen 1 Motivation Determinant is a function that each square real matrix A is assigned a real number, denoted det A, satisfying certain properties If A is a 3 3 matrix, writing A [u,
More informationENGR1100 Introduction to Engineering Analysis. Lecture 21
ENGR1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility
More informationLaplacian eigenvalues and optimality: II. The Laplacian of a graph. R. A. Bailey and Peter Cameron
Laplacian eigenvalues and optimality: II. The Laplacian of a graph R. A. Bailey and Peter Cameron London Taught Course Centre, June 2012 The Laplacian of a graph This lecture will be about the Laplacian
More informationThe third smallest eigenvalue of the Laplacian matrix
Electronic Journal of Linear Algebra Volume 8 ELA Volume 8 (001) Article 11 001 The third smallest eigenvalue of the Laplacian matrix Sukanta Pati pati@iitg.ernet.in Follow this and additional works at:
More informationStrongly Regular Decompositions of the Complete Graph
Journal of Algebraic Combinatorics, 17, 181 201, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Strongly Regular Decompositions of the Complete Graph EDWIN R. VAN DAM Edwin.vanDam@uvt.nl
More information. The following is a 3 3 orthogonal matrix: 2/3 1/3 2/3 2/3 2/3 1/3 1/3 2/3 2/3
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. An n n matrix
More informationChapter 4. Determinants
4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining
More informationTrees. A tree is a graph which is. (a) Connected and. (b) has no cycles (acyclic).
Trees A tree is a graph which is (a) Connected and (b) has no cycles (acyclic). 1 Lemma 1 Let the components of G be C 1, C 2,..., C r, Suppose e = (u, v) / E, u C i, v C j. (a) i = j ω(g + e) = ω(g).
More information5 Flows and cuts in digraphs
5 Flows and cuts in digraphs Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset of ordered pairs of elements of V, which we refer to as arcs. Note that two vertices
More informationMath 396. An application of GramSchmidt to prove connectedness
Math 396. An application of GramSchmidt to prove connectedness 1. Motivation and background Let V be an ndimensional vector space over R, and define GL(V ) to be the set of invertible linear maps V V
More informationA Characterization of (3+1)Free Posets
Journal of Combinatorial Theory, Series A 93, 231241 (2001) doi:10.1006jcta.2000.3075, available online at http:www.idealibrary.com on A Characterization of (3+1)Free Posets Mark Skandera Department of
More informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
More informationDeterminants. Recall that the 2 2 matrix a b c d. is invertible if
Determinants Recall that the 2 2 matrix a b c d is invertible if and only if the quantity ad bc is nonzero. Since this quantity helps to determine the invertibility of the matrix, we call it the determinant.
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationMath 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants
THE ALGEBRA OF DETERMINANTS 1. Determinants We have already defined the determinant of a 2 2 matrix: det = ad bc. We ve also seen that it s handy for determining when a matrix is invertible, and when it
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and
More informationMATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.
MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication
More information1 Positive definiteness and semidefiniteness
Positive definiteness and semidefiniteness Zdeněk Dvořák May 9, 205 For integers a, b, and c, let D(a, b, c) be the diagonal matrix with + for i =,..., a, D i,i = for i = a +,..., a + b,. 0 for i = a +
More information5.6. PSEUDOINVERSES 101. A H w.
5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the leastsquares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationThe complexity of counting graph homomorphisms
The complexity of counting graph homomorphisms Martin Dyer and Catherine Greenhill Abstract The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalisation of
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationDeterminants: Elementary Row/Column Operations
Determinants: Elementary Row/Column Operations Linear Algebra Josh Engwer TTU 23 September 2015 Josh Engwer (TTU) Determinants: Elementary Row/Column Operations 23 September 2015 1 / 16 Elementary Row
More informationThe Singular Acyclic Matrices of Even Order with a PSet of Maximum Size
Filomat 30:13 (016), 3403 3409 DOI 1098/FIL1613403D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat The Singular Acyclic Matrices of
More informationSpectrally arbitrary star sign patterns
Linear Algebra and its Applications 400 (2005) 99 119 wwwelseviercom/locate/laa Spectrally arbitrary star sign patterns G MacGillivray, RM Tifenbach, P van den Driessche Department of Mathematics and Statistics,
More informationMath 4377/6308 Advanced Linear Algebra
2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377
More informationLecture 13 Spectral Graph Algorithms
COMS 9953: Advanced Algorithms March 6, 7 Lecture 3 Spectral Graph Algorithms Instructor: Alex Andoni Scribe: Srikar Varadaraj Introduction Today s topics: Finish proof from last lecture Example of random
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems PerOlof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationOn the eigenvalues of Euclidean distance matrices
Volume 27, N. 3, pp. 237 250, 2008 Copyright 2008 SBMAC ISSN 008205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH Department of Mathematics and Statistics University
More informationLecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,
2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for
More informationMatrix Operations: Determinant
Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant
More informationFoundations of Cryptography
Foundations of Cryptography Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi Department of Mathematics and Statistics University of Turku 2017 Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi
More informationDOMINO TILING. Contents 1. Introduction 1 2. Rectangular Grids 2 Acknowledgments 10 References 10
DOMINO TILING KASPER BORYS Abstract In this paper we explore the problem of domino tiling: tessellating a region with x2 rectangular dominoes First we address the question of existence for domino tilings
More informationMATH 6102: PRACTICE PROBLEMS FOR FINAL EXAM
MATH 6102: PRACTICE PROBLEMS FOR FINAL EXAM (FP1) The exclusive or operation, denoted by and sometimes known as XOR, is defined so that P Q is true iff P is true or Q is true, but not both. Prove (through
More informationDeterminants. 2.1 Determinants by Cofactor Expansion. Recall from Theorem that the 2 2 matrix
CHAPTER 2 Determinants CHAPTER CONTENTS 21 Determinants by Cofactor Expansion 105 22 Evaluating Determinants by Row Reduction 113 23 Properties of Determinants; Cramer s Rule 118 INTRODUCTION In this chapter
More informationReducing the adjacency matrix of a tree
Electronic Journal of Linear Algebra Volume ELA Volume (996) Article 3 996 Reducing the adjacency matrix of a tree Gerd H. Fricke g.fricke@moreheadstate.edu Stephen T. Hedetniemi shedet@cs.clemson.edu
More informationSymmetric matrices and dot products
Symmetric matrices and dot products Proposition An n n matrix A is symmetric iff, for all x, y in R n, (Ax) y = x (Ay). Proof. If A is symmetric, then (Ax) y = x T A T y = x T Ay = x (Ay). If equality
More informationDeterminants An Introduction
Determinants An Introduction Professor Je rey Stuart Department of Mathematics Paci c Lutheran University Tacoma, WA 9844 USA je rey.stuart@plu.edu The determinant is a useful function that takes a square
More informationCRITICAL GROUPS AND LINE GRAPHS
CRITICAL GROUPS AND LINE GRAPHS ANDREW BERGET 1. Introduction This paper is an overview of what the author has learned about the critical group of a graph, including some new results. In particular we
More information5 Quiver Representations
5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )
More informationCluster algebras, snake graphs and continued fractions. Ralf Schiffler
Cluster algebras, snake graphs and continued fractions Ralf Schiffler Intro Cluster algebras Continued fractions Snake graphs Intro Cluster algebras Continued fractions expansion formula via perfect matchings
More informationGraph Theoretic Methods for Matrix Completion Problems
Graph Theoretic Methods for Matrix Completion Problems 1 Leslie Hogben Department of Mathematics Iowa State University Ames, IA 50011 lhogben@iastate.edu Abstract A pattern is a list of positions in an
More informationThe Adjacency Matrix, Standard Laplacian, and Normalized Laplacian, and Some Eigenvalue Interlacing Results
The Adjacency Matrix, Standard Laplacian, and Normalized Laplacian, and Some Eigenvalue Interlacing Results Frank J. Hall Department of Mathematics and Statistics Georgia State University Atlanta, GA 30303
More informationData Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings
Data Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings Radu Horaud INRIA Grenoble RhoneAlpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline
More information17.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs
Chapter 17 Graphs and Graph Laplacians 17.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs Definition 17.1. A directed graph isa pairg =(V,E), where V = {v
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188196. [SB], Chapter 26, p.719739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationOn Expected Gaussian Random Determinants
On Expected Gaussian Random Determinants Moo K. Chung 1 Department of Statistics University of WisconsinMadison 1210 West Dayton St. Madison, WI 53706 Abstract The expectation of random determinants whose
More information