 10.3.1E: Find real numbers a, b, and c such that the following are true.
 10.3.2E: Find the adjacency matrices for the following directed graphs.
 10.3.3E: Find directed graphs that have the following adjacency matrices:
 10.3.4E: Find adjacency matrices for the following (undirected) graphs. c. K...
 10.3.5E: Find graphs that have the following adjacency matrices.
 10.3.6E: The following are adjacency matrices for graphs. In each case deter...
 10.3.7E: Suppose that for all positive integers i , all the entries in the i...
 10.3.8E: Find each of the following products.
 10.3.9E: Find each of the following products.
 10.3.10E: For each of the following, determine whether the indicated product ...
 10.3.11E: Give an example different from that in the text to show that matrix...
 10.3.12E: Let O denote the matrix Find 2 × 2 matrices A and B such that A ? O...
 10.3.13E: Let O denote the matrix Find 2 × 2 matrices A and B such that A ? B...
 10.3.14E: In 14–18 assume the entries of all matrices are real numbers.Prove ...
 10.3.15E: In 14–18 assume the entries of all matrices are real numbers.Prove ...
 10.3.16E: In 14–18 assume the entries of all matrices are real numbers.Prove ...
 10.3.17E: In 14–18 assume the entries of all matrices are real numbers.Use ma...
 10.3.18E: In 14–18 assume the entries of all matrices are real numbers.Use ma...
 10.3.19E: b. Let G be the graph with vertices v1, v2, and v3 and with A as it...
 10.3.20E: The following is an adjacency matrix for a graph: Answer the follow...
 10.3.21E: Let A be the adjacent matrix for K3, the complete graph on three ve...
 10.3.22E: a. Draw a graph that has as its adjacency matrix. Is this graph bip...
 10.3.23E: a. Let G be a graphwith n vertices, and let v and w be distinct ver...
Solutions for Chapter 10.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.3
Get Full SolutionsChapter 10.3 includes 23 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Since 23 problems in chapter 10.3 have been answered, more than 24064 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Lucas numbers
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.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).