 10.3.1: Find real numbers a, b, and c such that the following are true. a. ...
 10.3.2: Find the adjacency matrices for the following directed graphs. a. v...
 10.3.3: Find directed graphs that have the following adjacency matrices:
 10.3.4: Find adjacency matrices for the following (undirected) graphs. v1 v...
 10.3.5: Find graphs that have the following adjacency matrices
 10.3.6: The following are adjacency matrices for graphs. In each case deter...
 10.3.7: Suppose that for all positive integers i, all the entries in the it...
 10.3.8: Find each of the following product
 10.3.9: Find each of the following products.
 10.3.10: Let A = * 1 1 1 0 2 1+ , B = * 2 0 1 3+ , and C = 0 2 3 1 1 0 . For...
 10.3.11: Give an example different from that in the text to show that matrix...
 10.3.12: Let O denote the matrix * 0 0 0 0+ . Find 2 2 matrices A and B such...
 10.3.13: Let O denote the matrix * 0 0 0 0+ . Find 2 2 matrices A and B such...
 10.3.14: In 1418 assume the entries of all matrices are real numbers.
 10.3.15: In 1418 assume the entries of all matrices are real numbers.
 10.3.16: In 1418 assume the entries of all matrices are real numbers.
 10.3.17: In 1418 assume the entries of all matrices are real numbers.
 10.3.18: In 1418 assume the entries of all matrices are real numbers.
 10.3.19: a. Let A = 112 101 210 . Find A2 and A3 . b. Let G be the graph wit...
 10.3.20: The following is an adjacency matrix for a graph: v1 v2 v3 v4 v1 01...
 10.3.21: Let A be the adjacent matrix for K3, the complete graph on three ve...
 10.3.22: a. Draw a graph that has 00012 00011 00021 11200 21100 as its adjac...
 10.3.23: a. Let G be a graph with n vertices, and let v and w be distinct ve...
Solutions for Chapter 10.3: Matrix Representations of Graphs
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.3: Matrix Representations of Graphs
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 10.3: Matrix Representations of Graphs includes 23 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 23 problems in chapter 10.3: Matrix Representations of Graphs have been answered, more than 45392 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.