# The Symmetric M-Matrix and Symmetric Inverse M-Matrix Completion Problems

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The Symmetric M-Matrix and Symmetric Inverse M-Matrix Completion Problems Leslie Hogben Department of Mathematics Iowa State University Ames, IA Abstract The symmetric M-matrix and symmetric M 0 -matrix completion problems are solved and results of Johnson and Smith [JS2] are extended to solve the symmetric inverse M-matrix completion problem: 1) A pattern (i.e., a list of positions in an n n matrix) has symmetric M- completion (i.e., every partial symmetric M-matrix specifying the pattern can be completed to a symmetric M-matrix) if and only if the principal subpattern R determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete, or, in graph theoretic terms, if and only if each component of the graph of R is a clique. 2) A pattern has symmetric M 0 -completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions, or, in graph theoretic terms, if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions, or includes all positions. 3) A pattern has symmetric inverse M-completion if and only if its graph is block-clique and no diagonal position is omitted that corresponds to a vertex in a graph-block of order > 2. The techniques used are also applied to matrix completion problems for other classes of symmetric matrices. 1. Introduction A partial matrix is a matrix in which some entries are specified and others are not. A completion of a partial matrix is a matrix obtained by choosing values for the unspecified entries. A pattern for n n matrices is a list of positions of an n n matrix, that is, a subset of {1,...,n} {1,...,n}. A partial matrix specifies a pattern if its specified positions are exactly those listed in the pattern. Note that in this paper a pattern does not need to include all diagonal positions. All matrices and partial matrices discussed here are real. The symbol Π will denote a class of matrices and a Π-matrix is a matrix in the class Π. For a particular class Π of matrices, the Π- matrix completion problem for patterns asks which patterns have the property that any partial Π- matrix that specifies the pattern can be completed to a Π-matrix. When a pattern has this property, we say it has Π-completion. The answer to the Π-matrix completion problem obviously depends on the definition of partial Π-matrix. For many classes Π of matrices, in order for it to be possible to have a completion of a partial matrix to a Π-matrix, certain obviously necessary conditions must be satisfied. Such obviously necessary conditions are frequently taken as the definition of a partial Π-matrix, [JS1], [JS2], [H1], [H2], [H3], [H4]. Here we also take this approach of using obviously necessary conditions to define a partial Π-matrix (see below). In this paper we concern ourselves only with matrix completion problems for patterns. Such problems have been studied for M-matrices [H2], M 0 -matrices [H4], inverse M-matrices 1

2 [JS1], [H1], [H3], symmetric inverse M-matrices [JS2], and many other classes. Results on matrix completion problems and techniques are surveyed in [H4]. For α a subset of {1,...,n}, the principal submatrix A[α] is obtained from the n n matrix A by deleting all entries a ij such that i α or j α. Similarly, the principal subpattern Q[α] = Q (α α). The principal subpattern determined by the diagonal positions is Q[δ] where δ = {i (i,i) Q}. The characteristic matrix of a pattern Q for n n matrices is the n n matrix C Q such that c ij = 1 if the position (i,j) is in the pattern and c ij = 0 if (i,j) is not in the pattern. A pattern Q is permutation similar to a pattern R if C Q is permutation similar to C R. A pattern is block diagonal (for a particular block structure) if its characteristic matrix is block diagonal for that block structure (cf. Section 3 of [H4]). A class Π of matrices is called a Hereditary-Sum-Permutation-closed (HSP) class if (1) every principal submatrix of a Π-matrix is a Π-matrix, (2) the direct sum of Π-matrices is a Π- matrix, (3) if A is a Π-matrix and P is a permutation matrix of the same size then PAP -1 is a Π- matrix, and (4) there is a 1 1 Π-matrix. All of the classes discussed in [H4], except those that require entries to be positive (and thus fail condition (2)), are HSP classes. For an HSP class Π we frequently define a partial matrix B to be a partial Π-matrix if any fully specified principal submatrix of B is a Π-matrix and any sign condition on the entries of a Π-matrix is respected by B (here sign condition includes nonpositive, nonnegative, sign symmetric or weakly sign symmetric, cf. [H4]). Using this definition of a partial Π-matrix is referred to as using the HSP standard definition of a partial Π-matrix. A class Σ of matrices is called symmetric if every Σ-matrix is symmetric. A partial matrix B is symmetric if whenever b ij is specified then so is b ji and b ji = b ij. A pattern is symmetric (also called positionally symmetric and combinatorially symmetric ) if position (i,j) in the pattern implies (j,i) is also in the pattern. A class Σ of matrices is called a Symmetric-Hereditary-Sum- Permutation-closed (SHSP) class if Σ is both symmetric and an HSP class. For an SHSP class Σ, we frequently define a partial Σ -matrix to be a symmetric partial matrix meeting the requirements for a partial matrix of an HSP class. Using this definition of a partial Π-matrix is referred to as using the SHSP standard definition of partial Π-matrix. The matrix A is called positive stable (respectively, semistable) if all the eigenvalues of A have positive (nonnegative) real part. An M-matrix (respectively, M 0 -matrix) is a positive stable (semistable) matrix with nonpositive off-diagonal entries. There are many equivalent characterizations of M- and M 0 -matrices [HJ]: A matrix with nonpositive off-diagonal entries is an M-matrix (M 0 -matrix) if and only if every principal minor is positive (nonnegative). A matrix with nonpositive off-diagonal entries is an M-matrix if and only if it is nonsingular and its inverse is entrywise nonnegative. The notation M (0) will be used to mean M (respectively, M 0 ). The matrix B is an inverse M-matrix if B is the inverse of an M-matrix. Equivalently, an inverse M-matrix is a nonsingular, entrywise nonnegative matrix B such that B -1 has nonpositive off-diagonal entries. A substantial amount is known about M-matrices, M 0 -matrices and inverse M-matrices [HJ], [J], [LN], including the fact that each of these classes is an HSP class. Use the HSP standard definition of a partial Π-matrix: A partial M (0) -matrix is a partial matrix such that any fully specified principal submatrix is an M (0) -matrix and all specified offdiagonal entries are nonpositive. A partial inverse M-matrix is an entrywise nonnegative partial matrix such that any fully specified principal submatrix is an inverse M-matrix. We will follow the notation of [JS2] in referring to a symmetric inverse M-matrix as a SIM matrix and we refer to a symmetric M (0) - matrix as a SM (0) -matrix. Note that the three classes SIM, SM and SM 0 are SHSP classes. Use the SHSP standard definition of a partial Π-matrix: A partial SIM-matrix is a partial inverse M-matrix that is symmetric. A partial SM (0) - matrix is a partial M (0) -matrix that is symmetric. 2

3 With the HSP and SHSP standard definitions of partial Π-matrix, HSP and SHSP classes have two basic properties that are used extensively in the study of matrix completion problems, Lemma 1.1 and Observation 1.2 below. 1.1 Lemma Let Π be an HSP (SHSP) class using the HSP (SHSP) standard definition of a partial Π-matrix. If the pattern Q has Π-completion then so does every principal subpattern Q[α]. Proof: Let A be a partial Π-matrix specifying Q[α]. By (4), there is some 1 1 Π-matrix [s]. Extend A to a partial matrix B specifying Q by, for each (i,j) in Q but not in Q[α], setting b ij = s if i = j and 0 otherwise. Then B is a partial Π-matrix because any fully specified principal submatrix B[β] of B is permutation similar to A[α β] [s]... [s], which is a Π-matrix by (2). So B[β] is a Π-matrix by (3). By hypothesis, B can be completed to a Π-matrix C. Then C[α], which completes A, is a Π-matrix by (1). Our graph terminology follows [H4]. For a symmetric pattern Q, the pattern-graph of Q is the graph having {1,...,n} as its vertex set and, as its set of edges, the set of (unordered) pairs {i,j} such that position (i,j) (and therefore also (j,i)) is in Q. If G is the pattern-graph of Q, then the pattern-graph of a principal subpattern Q[α] is <α>, the subgraph induced by α. The principal subpattern Q[α] and the induced subgraph <α> are said to correspond; in particular, the vertex v and diagonal position (v,v) correspond. Renaming the vertices of a pattern-graph is equivalent to applying a permutation similarity to the pattern. A component of a graph is a maximal connected subgraph. A cut-vertex of a connected graph is a vertex whose deletion disconnects the graph; more generally, a cut-vertex is a vertex whose deletion disconnects the component containing it. A graph is nonseparable if it is connected and has no cut-vertices. A block of a graph is a subgraph that is nonseparable and is maximal with respect to this property. A (sub)graph is called a clique if it contains all possible edges between its vertices. A graph is block-clique if every block is a clique. Block-clique graphs are called 1 chordal in [JS2]. For matrices and patterns that need not be symmetric, digraphs must be used. Let A be a (fully specified) n n matrix. The nonzero-digraph of A is the digraph having as vertex set {1,...,n}, and, as its set of arcs, the set of ordered pairs (i,j) such that both i and j are vertices with i j and a ij 0. For a pattern Q that need not be symmetric, the pattern-digraph of Q is the digraph having {1,...,n} as its vertex set and members (i,j) of Q with i j as its arcs. A digraph is transitive if the existence of a path from v to w implies the arc (v,w) is in the digraph. Recall that the nonzero-digraph of any inverse M-matrix is transitive [LN]. 1.2 Observation Let Π is an HSP (SHSP) class. If the pattern Q is permutation similar to a block diagonal pattern in which each diagonal block has Π-completion, then Q has Π-completion by (2) and (3). Equivalently, if each principal subpattern of Q corresponding to a component of a patterndigraph (pattern-graph) has Π-completion, then the pattern has Π-completion. The results in 1.1 and 1.2 are already known for SM (0) -matrices and SIM-matrices [H4]. Johnson and Smith [JS2] determined that a symmetric pattern that includes all diagonal positions has SIM completion if and only if its pattern-graph is block-clique. More general patterns that may omit some diagonal positions are classified as to SIM completion in the next section. All patterns are classified as to SM (0) -completion in the third section. 2. Determination of patterns having SIM completion It is well known that a graph G is block-clique if and only if for every cycle v 1, v 2..., v k, v 1 of G, the induced subgraph <{v 1, v 2..., v k }> is a clique [JS1]. 3

4 2.1 Theorem Let Q be a symmetric pattern and let G be its pattern-graph. If Q has SIM completion, then G is block-clique and the diagonal positions corresponding to the vertices of every cycle in G are all included in Q. Proof: Suppose Q and G do not have the required property. Then G is contains a cycle whose induced subgraph is not a clique or Q omits the diagonal position corresponding to a vertex in a cycle. Let Γ be a shortest troublesome cycle. By renaming vertices if necessary, assume Γ = 1, 2,..., k, 1 with k > 2, and <{1,,k}> is not a clique or diagonal position (k,k) is not in Q. Suppose first that <{1,...,k}> does not contain any chord of Γ. If k > 3, then Q[{1,...,k}] does not contain any complete principal subpattern of size larger than 2 2, because all chords are omitted. When k = 3, (3,3) must be omitted from Q. Thus, in either case, Q[{1,...,k}] does not contain any complete principal subpattern of size larger than 2 2. Define a k k partial matrix B specifying Q[{1,..,k}] by setting b ii = 2 for (i,i) Q[{1,..,k}], setting b i i+1 = 1 = b i+1 i for i=1,...,k- 1, and setting all other specified entries (including b 1k and b k1 ) equal to 0. Since the only completely specified principal submatrices are 2 2 or smaller, and these are SIM matrices, B is a partial SIM matrix. But B cannot be completed to a SIM matrix because the nonzero-digraph of any completion of B is not transitive, since b 12 =... = b k-1 k = 1 and b 1k = 0. So Q[{1,...,k}] does not have SIM completion. Now suppose k > 3 and <{1,...,k}> contains a chord of Γ. Each of the two pieces of Γ on either side of the chord, together with the chord, forms a shorter cycle. By the minimal length assumption, Q must include all diagonal positions corresponding to vertices in these two shorter cycles. Hence, Q includes all diagonal positions (1,1),..., (k,k). Then <{1,..,k}> is not a clique, and thus is not block-clique. So Q[{1,...,k}] is a symmetric pattern that includes all diagonal positions and whose graph is not block-clique, and therefore does not have SIM completion [JS2]. In either case Q does not have SIM completion because Q[{1,...,k}] does not. The properties that for any vertex v that appears in a cycle of G, (v,v) Q, and the induced subgraph of a cycle of G must be a clique are the graph analog of the pattern-digraph condition path-clique, which is necessary for a (not necessarily symmetric) pattern to have inverse M completion [H3]. (In a digraph D, an alternate path to a single arc is a path (v 1,v 2 ), (v 2,v 3 ),..., (v k- 1,v k ) with k > 2, such that (v 1,v k ) is an arc of D. A pattern-digraph is called path-clique if the induced subdigraph of any alternate path to a single arc is a clique and the diagonal position (v i,v i ) is in the pattern for every vertex v i in the path). 2.2 Theorem A symmetric pattern Q has SIM completion if and only if its pattern-graph G is block-clique and the diagonal position (v,v) is in Q for every vertex v in a block of order > 2. Proof: (only if) By Theorem 2.1, G is block-clique. Let v be a vertex in a block H of order > 2. There are two other distinct vertices u and w in H. Since H is a clique, {v,u}, {u,w} and {w,v} are in H, and v occurs in a cycle. So by Theorem 2.1 (v,v) is in Q. (if) Let H be a block of G of order > 2. By hypothesis, H is a clique and (v,v) is in Q for every vertex v in H. Thus the principal subpattern Q[η] corresponding to H contains all positions, so Q[η] trivially has SIM-completion. Any symmetric pattern for 2 2 matrices has SIM completion [H4, remark following Lemma 4.8], so the principal subpattern of Q corresponding to any block of G of order 2 has SIM completion. Thus the principal subpattern of Q corresponding to each block of G is has SIM completion, and so Q has SIM completion by Corollary 5.6 of [H4]. The following example exhibits patterns that do not include all diagonal positions, one having SIM completion and other not having SIM completion. In the diagrams of the patterngraphs, if a diagonal position (v,v) is in the pattern, then vertex v is indicated by a solid black dot ( ); if (v,v) is omitted, then vertex v is indicated by a hollow circle ( ) (this follows the notation of [H3] and [H4]). 4

5 Figure 1 (a) Q 1 has SIM completion (b) Q 2 does not have SIM completion 2.3 Example The pattern Q 1 = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (4,7), (5,3), (5,4), (5,5), (5,6), (5,8), (6,3), (6,4), (6,5), (6,6), (7,4), (8,5), (8,9), (8,11), (9,8), (9,10), (10,9), (10,10), (11,8), (11,11), (11,12), (11,13), (12,11), (12,12), (12,13), (13,11), (13,12), (13,13)}, whose pattern-graph is shown in Figure 1(a), has SIM completion. The pattern Q 2 obtained from Q 1 by deleting the diagonal position (3,3), whose pattern-graph is shown in Figure 1(b), does not. 3. Determination of patterns having SM (0) -completion A partial M (0) -matrix with all diagonal entries specified can be completed to an M (0) -matrix if only if its zero completion (i.e., the result of setting all unspecified entries to 0) is an M (0) -matrix, cf. [JS1], [H4]. Since a partial SM (0) -matrix is an M (0) -matrix, and the zero completion of a partial SM (0) -matrix is symmetric, a partial SM (0) -matrix with all diagonal entries specified can be completed to an SM (0) -matrix if only if its zero completion is an SM (0) -matrix. 3.1 Lemma If a symmetric pattern Q has SM (0) - completion and includes positions (a,a), (b,b), (c,c), (a,b), (b,a), (b,c), and (c,b) with a<b<c, then Q also includes (a,c) and (c,a). Proof: Suppose Q does not include (a,c) and (c,a). Then the partial matrix SM-matrix 4 3? A = specifies Q[{a,b,c}] and cannot be completed to an SM 0 -matrix because the? 3 4 zero-completion of A has determinant -8. Thus Q does not have SM (0) - completion. 3.2 Theorem Let Q be a symmetric pattern with the property that if (v,w) is in Q then (v,v) and (w,w) are both in Q. Then the following are equivalent: 1) Q has SM (0) -completion. 2) Q is permutation similar to a block diagonal pattern with each diagonal block containing all positions or consisting of a single omitted diagonal position. 3) Each component of its pattern-graph is a clique. Proof: The equivalence of the (2) and (3) is immediate from the hypothesis about Q. If Q satisfies (3), then Q has SM (0) -completion by Observation 1.2. Let Q have SM (0) -completion. Let u and v be vertices of H, a component of order > 1 of the pattern-graph of Q. Since (w,w) is in Q for every vertex visited by a path in H connecting u and v, we may apply Lemma 3.1 to eliminate one at a time from that path any vertex other than u and v. Hence {v,u} is in H and H is a clique. 5

6 A symmetric pattern Q has SM-completion if and only if Q[δ] does [H4, Theorem 4.6]. The principal subpattern Q[δ] satisfies the hypotheses of Theorem 3.2, so this completes the determination of patterns having SM-completion. 3.3 Corollary A symmetric pattern Q has SM-completion if and only if Q[δ] is permutation similar to a pattern that is block diagonal with all positions in each of the blocks on the diagonal in the pattern; in graph theoretic language, if and only if each component of the pattern-graph of Q[δ] is a clique. Figure 2 (a) Q 1 [δ] (b) Q 2 [δ] 3.4 Example The pattern Q 1 in Example 2.3 does not have SM-completion because one of the components of the pattern-graph of Q 1 [δ] is not a clique. The pattern Q 2 in Example 2.3 has SMcompletion, because each component of Q 2 [δ] is a clique. The pattern-graphs of Q 1 [δ] and Q 2 [δ] are shown in Figure 2(a) and (b), respectively. 3.5 Example The partial SM 0 -matrix A = 0 1 1? cannot be completed to an SM 0-matrix because the determinant of any completion of A equals -1. Thus, neither Q 1 nor Q 2 from Example 2.3 has SM 0 -completion, because both contain the principal subpattern R = {(4,4), (4,7), (7,4)}. 3.6 Lemma Let Π be an HSP (SHSP) class (using the HSP (SHSP) standard definition of a partial Π-matrix) such that {(a,a), (a,b), (b,a)} (with a b) does not have Π-completion. Then if a symmetric pattern Q has Π-completion, Q is permutation similar to a block diagonal pattern in which every diagonal block either includes all diagonal positions or omits all diagonal positions, i. e., in graph theoretic terms, every principal subpattern R corresponding to a component H of the pattern-graph G of Q includes all diagonal positions or omits all diagonal positions. Proof: Suppose R includes (v,v) and omits (w,w). Since H is a component, it is connected, and it contains a path {u 1,u 2 }, {u 2,u 3 },..., {u k-1,u k } from vertex v = u 1 to vertex w = u k. Let t be the number such that R includes (u 1, u 1 ),..., (u t, u t ) and R does not include (u t+1, u t+1 ). Then Q[{u t, u t+1 }] = {(u t, u t ),( u t, u t+1 ),( u t+1,u t )}. Thus by the hypothesis, Q[{u t, u t+1 }] does not have Π-completion, and so neither does Q, by Lemma Theorem Let Q be a symmetric pattern and let G be its pattern-graph. Then Q has SM 0 - completion if and only if Q is permutation similar to a pattern that is block diagonal in which each diagonal block either omits all diagonal positions or includes all positions, i. e., in graph theoretic language, if and only if every principal subpattern corresponding to a component of G either omits all diagonal positions, or includes all positions. 6

7 Proof: If Q has SM 0 -completion, 3.6 shows that after permutation similarity the diagonal blocks have either no or all diagonal positions. Theorem 3.2 shows that in the latter case the block includes all positions. Conversely, a pattern that omits all diagonal positions has SM 0 -completion [H4, Theorem 4.7]. A pattern that includes all positions trivially has SM 0 -completion. Since Q is permutation similar to a block-diagonal pattern in which each diagonal block has SM 0 -completion, Q has SM 0 -completion by Corollary Any symmetric pattern that has SM 0 -completion also has SM-completion (cf. 3.3), but the converse is false (cf. 3.5). A symmetric pattern that includes all diagonal positions and has SM-completion also has SM 0 -completion (cf. 3.2). The next example shows that the assumption in Lemma 3.6 that the pattern is symmetric is necessary. A matrix is a P 0 -matrix if every principal minor is non-negative. Use the HSP standard definition of a partial Π-matrix: A partial P 0 -matrix is a partial matrix such that any fully specified principal submatrix is an P 0 -matrix. The pattern {(a,a), (a,b), (b,a)} (with a b) does not have P 0 - completion (cf. Example 3.5), so Lemma 3.6 applies to the class of P 0 -matrices. 3.9 Example The pattern {(1,1), (1,2), (2,2), (2,3), (3,1)} (whose pattern-digraph is a 3-cycle) has P 0 -completion, because it is asymmetric [CDHMW], and neither contains nor omits all diagonal positions. We can also use Lemma 3.6 to complete the classification of patterns for other classes of symmetric matrices. The matrix A is doubly nonnegative (DN) if A is entrywise nonnegative and positive semidefinite. The matrix A is completely positive (CP) if A is entrywise nonnegative and A = BB T for some entrywise nonnegative n m matrix B (the requirement that A be entrywise nonnegative is clearly redundant, but helps clarify the interpretation of using the SHSP standard definition of a partial CP-matrix). The classes DN and CP are SHSP classes [DJ]. Use the SHSP standard definitions of a partial Π-matrix: A partial DN-matrix is an entrywise nonnegative symmetric partial matrix such that any fully specified principal submatrix is a DN-matrix. A partial CP-matrix is an entrywise nonnegative symmetric partial matrix such that any fully specified principal submatrix is a CP-matrix. Drew and Johnson [DJ] established that a symmetric pattern that includes the diagonal has DN- (CP-) completion if and only if its pattern-graph is block-clique. 0 1 The partial matrix 1? shows that {(a,a), (a,b), (b,a)} does not have DN- or CP-completion. Since any diagonally dominant nonnegative symmetric matrix is CP [K] (and thus DN), a pattern that omits all diagonal positions has CP- and DN-completion Corollary Let Q be a symmetric pattern and let G be its pattern-graph. Then Q has DN- (CP-) completion if and only if every principal subpattern corresponding to a component H of G either omits all diagonal positions, or includes all diagonal positions and H is block-clique. References [CDHMW] [DJ] Choi, DeAlba, Hogben, Maxwell, and Wangsness, The P o - Matrix Completion Problem, to appear in Electronic Journal of Linear Algebra. J. H. Drew and C. R. Johnson, The Completely Positive and Doubly Nonnegative Completion Problems, Linear and Multilinear Algebra 44 (1998),

8 [H1] [H2] [H3] [H4] [HJ] L. Hogben, Completions of Inverse M-matrix Patterns. Linear Algebra and Its Applications 282 (1998), L. Hogben, Completions of M-matrix Patterns. Linear Algebra and Its Applications 285 (1998), L. Hogben, Inverse M-matrix Completions of Patterns Omitting Some Diagonal Positions, Linear Algebra and Its Applications 313 (2000), L. Hogben, Graph Theoretic Methods for Matrix Completion Problems, Linear Algebra and Its Applications 328 (2001) R. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [J] C. R. Johnson, Inverse M-Matrices, Linear Algebra and Its Applications 47 (1982), [JS1] [JS2] [K] [LN] C. R. Johnson and R. L. Smith, The Completion Problem for M-Matrices and Inverse M-matrices, Linear Algebra and Its Applications (1996), C. R. Johnson and R. L. Smith, The Symmetric Inverse M-Matrix Completion Problem, Linear Algebra and Its Applications 290 (1999), M. Kaykobad, On Nonnegative Factorization of Matrices, Linear Algebra and Its Applications 96 (1987), M. Lewin and M. Neumann, On the Inverse M-matrix Problem for (0,1)-Matrices, Linear Algebra and Its Applications 30 (1980):

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### ELA http://math.technion.ac.il/iic/ela

SIGN PATTERNS THAT ALLOW EVENTUAL POSITIVITY ABRAHAM BERMAN, MINERVA CATRAL, LUZ M. DEALBA, ABED ELHASHASH, FRANK J. HALL, LESLIE HOGBEN, IN-JAE KIM, D. D. OLESKY, PABLO TARAZAGA, MICHAEL J. TSATSOMEROS,

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### A note on companion matrices

Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod

### Chapter 5. Matrices. 5.1 Inverses, Part 1

Chapter 5 Matrices The classification result at the end of the previous chapter says that any finite-dimensional vector space looks like a space of column vectors. In the next couple of chapters we re

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### DETERMINANTS 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 RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

### Suk-Geun Hwang and Jin-Woo Park

Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGN-SOLVABILITY Suk-Geun Hwang and Jin-Woo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### 2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

### Matrices, transposes, and inverses

Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

### ELA

BLOCK REPRESENTATIONS OF THE DRAZIN INVERSE OF A BIPARTITE MATRIX M. CATRAL, D.D. OLESKY, AND P. VAN DEN DRIESSCHE 2 Abstract. Block representations of the Drazin inverse of a bipartite matrix A = 4 0

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### Define the set of rational numbers to be the set of equivalence classes under #.

Rational Numbers There are four standard arithmetic operations: addition, subtraction, multiplication, and division. Just as we took differences of natural numbers to represent integers, here the essence

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Minimum rank of graphs that allow loops. Rana Catherine Mikkelson. A dissertation submitted to the graduate faculty

Minimum rank of graphs that allow loops by Rana Catherine Mikkelson A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major:

### EC9A0: Pre-sessional Advanced Mathematics Course

University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

### = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### UNCOUPLING THE PERRON EIGENVECTOR PROBLEM

UNCOUPLING THE PERRON EIGENVECTOR PROBLEM Carl D Meyer INTRODUCTION Foranonnegative irreducible matrix m m with spectral radius ρ,afundamental problem concerns the determination of the unique normalized

### TREES IN SET THEORY SPENCER UNGER

TREES IN SET THEORY SPENCER UNGER 1. Introduction Although I was unaware of it while writing the first two talks, my talks in the graduate student seminar have formed a coherent series. This talk can be

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?

MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All

### Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

### Matrix Inverse and Determinants

DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

### INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

### Solutions for Chapter 8

Solutions for Chapter 8 Solutions for exercises in section 8. 2 8.2.1. The eigenvalues are σ (A ={12, 6} with alg mult A (6 = 2, and it s clear that 12 = ρ(a σ (A. The eigenspace N(A 12I is spanned by

### Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### 13 Solutions for Section 6

13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution

### Algebraic and Combinatorial Circuits

C H A P T E R Algebraic and Combinatorial Circuits Algebraic circuits combine operations drawn from an algebraic system. In this chapter we develop algebraic and combinatorial circuits for a variety of

### The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

### Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices

Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 6 2010 Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices Roger A. Horn rhorn@math.utah.edu

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Synthetic Projective Treatment of Cevian Nests and Graves Triangles

Synthetic Projective Treatment of Cevian Nests and Graves Triangles Igor Minevich 1 Introduction Several proofs of the cevian nest theorem (given below) are known, including one using ratios along sides

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

FPSAC 2009 DMTCS proc (subm), by the authors, 1 10 Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph Christopher R H Hanusa 1 and Thomas Zaslavsky 2 1 Department

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### SECTIONS 1.5-1.6 NOTES ON GRAPH THEORY NOTATION AND ITS USE IN THE STUDY OF SPARSE SYMMETRIC MATRICES

SECIONS.5-.6 NOES ON GRPH HEORY NOION ND IS USE IN HE SUDY OF SPRSE SYMMERIC MRICES graph G ( X, E) consists of a finite set of nodes or vertices X and edges E. EXMPLE : road map of part of British Columbia

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

### ON THE TREE GRAPH OF A CONNECTED GRAPH

Discussiones Mathematicae Graph Theory 28 (2008 ) 501 510 ON THE TREE GRAPH OF A CONNECTED GRAPH Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria,

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### The tree-number and determinant expansions (Biggs 6-7)

The tree-number and determinant expansions (Biggs 6-7) André Schumacher March 20, 2006 Overview Biggs 6-7 [1] The tree-number κ(γ) κ(γ) and the Laplacian matrix The σ function Elementary (sub)graphs Coefficients

### Notes on spherical geometry

Notes on spherical geometry Math 130 Course web site: www.courses.fas.harvard.edu/5811 This handout covers some spherical trigonometry (following yan s exposition) and the topic of dual triangles. 1. Some

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from

4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called