 6.6.3E: The name of a file in a computer directory consists of three upperc...
 6.6.1E: Place these permutations of {1,2,3,4,5} in lexicographic order: 435...
 6.6.2E: Place these permutations of (1,2,3,4,5,6) in lexicographic order: 2...
 6.6.5E: Find the next larger permutation in lexicographic order after each ...
 6.6.4E: Suppose that the name of a file in a computer directory consists of...
 6.6.6E: Find the next larger permutation in lexicographic order after each ...
 6.6.7E: Use Algorithm 1 to generate the 24 permutations of the first four p...
 6.6.8E: Use Algorithm 2 to list all the subsets of the set {1, 2, 3, 4}.
 6.6.11E: Show that Algorithm 3 produces the next larger rcombination in lex...
 6.6.9E: Use Algorithm 3 to list all the 3combinations of {1,2,3,4,5}.
 6.6.12E: Develop an algorithm for generating the rpermutations of a set of ...
 6.6.10E: Show that Algorithm 1 produces the next larger permutation in lexic...
 6.6.13E: List all 3 permutations of {1, 2, 3, 4, 5}.The remaining exercises ...
 6.6.14E: Find the Cantor digits a1, a2,…, an?1 that correspond to these perm...
 6.6.15E: Show that the correspondence described in the preamble is a bijecti...
 6.6.16E: Find the permutations of {1, 2, 3, 4, 5} that correspond to these i...
 6.6.17E: Develop an algorithm for producing all permutations of a set of n e...
Solutions for Chapter 6.6: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 6.6
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Block matrix.
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.)

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

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).

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·