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Solutions for Chapter 52: Coloring

Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Solutions for Chapter 52: Coloring

Solutions for Chapter 52
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ISBN: 9780840049421

This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Chapter 52: Coloring includes 18 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 52: Coloring have been answered, more than 8988 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of 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.

• 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.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• 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.

• Linear combination cv + d w or L C jV j.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• 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 subspaces.

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

• Pascal matrix

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).

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Vector v in Rn.

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

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