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# Solutions for Chapter 12.2: Discrete Mathematics with Applications 4th Edition

## Full solutions for Discrete Mathematics with Applications | 4th Edition

ISBN: 9780495391326

Solutions for Chapter 12.2

Solutions for Chapter 12.2
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##### ISBN: 9780495391326

Chapter 12.2 includes 41 full step-by-step solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 12.2 have been answered, more than 52191 students have viewed full step-by-step solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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).

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• 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.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• 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.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• 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 .

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Orthogonal matrix Q.

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

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Skew-symmetric matrix K.

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

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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