- 10.1.10E: For each undirected graph in Exercises 3–9 that is not simple, find...
- 10.1.11E: Let G be a simple graph. Show that the relation R on the set of ver...
- 10.1.1E: Draw graph models, stating the type of graph (from Table 1) used, t...
- 10.1.2E: What kind of graph (from Table 1) can be used to model a highway sy...
- 10.1.14E: Use the niche overlap graph in Figure 11 to determine the species t...
- 10.1.13E: The intersection graph of a collection of sets A1, A2, .... An is t...
- 10.1.12E: Let G be an undirected graph with a loop at every vertex. Show that...
- 10.1.16E: Draw the acquaintanceship graph that represents that Tom and Patric...
- 10.1.15E: Construct a niche overlap graph for six species of birds, where the...
- 10.1.17E: We can use a graph to represent whether two people were alive at th...
- 10.1.18E: Who can influence Fred and whom can Fred influence in the influence...
- 10.1.19E: Construct an influence graph for the board members of a company if ...
- 10.1.21E: In a round-robin tournament the Tigers beat the Blue Jays, the Tige...
- 10.1.20E: Which other teams did Team 4 beat and which teams beat Team 4 in th...
- 10.1.22E: Construct the call graph for a set of seven telephone numbers 555–0...
- 10.1.25E: How can a graph that models e-mail messages sent in a network be us...
- 10.1.26E: How can a graph that models e-mail messages sent in a network be us...
- 10.1.24E: a) Explain how graphs can be used to model electronic mail messages...
- 10.1.23E: Explain how the two telephone call graphs for calls made during the...
- 10.1.28E: Describe a graph model that represents a subway system in a large c...
- 10.1.29E: For each course at a university, there may be one or more other cou...
- 10.1.27E: Describe a graph model that represents whether each person at a par...
- 10.1.30E: Describe a graph model that represents the positive recommendations...
- 10.1.31E: Describe a graph model that represents traditional marriages betwee...
- 10.1.32E: Which statements must be executed before S(, is executed in the pro...
- 10.1.34E: Describe a discrete structure based on a graph that can be used to ...
- 10.1.33E: Construct a precedence graph for the following program:
- 10.1.36E: Describe a graph model that can be used to represent all forms of e...
- 10.1.35E: Describe a discrete structure based on a graph that can be used to ...
Solutions for Chapter 10.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.