 15.1: Express each of the rational numbers below as finite simple continu...
 15.2: Determine the rational numbers represented by the following simple ...
 15.3: If r = [ao; ai, az, ... , an], where r > 1, show that1 = [O;ao, a1...
 15.4: Represent the following simple continued fractions in an equivalent...
 15.5: Compute the convergents of the following simple continued fractions...
 15.6: (a) If Ck = Pk/qk denotes the kth convergent of the finite simple c...
 15.7: Evaluate Pb qko and Ck(k = 0, 1, ... , 8) for the simple continued ...
 15.8: If Ck = Pk/qk is the kth convergent of the simple continued fractio...
 15.9: Find the simple continued fraction representation of 3.1416, and th...
 15.10: If Ck = Pk/qk is the kth convergent of the simple continued fractio...
 15.11: By means of continued fractions determine the general solutions of ...
 15.12: Assume that the continued fraction representation for the irrationa...
 15.13: Let x be an irrational number with convergents Pnf qn. For every n ...
 15.14: Apply the theory of this section to confirm that there exist infini...
 15.15: The Pell numbers Pn and qn are defined byPo= 0qo = 1P1 = 1qi= 1Pn =...
 15.16: For the Pell numbers, derive the relations below, where n :'.'.: 1:...
Solutions for Chapter 15: FINITE CONTINUED FRACTIONS
Full solutions for Elementary Number Theory  7th Edition
ISBN: 9780073383149
Solutions for Chapter 15: FINITE CONTINUED FRACTIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7. Chapter 15: FINITE CONTINUED FRACTIONS includes 16 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since 16 problems in chapter 15: FINITE CONTINUED FRACTIONS have been answered, more than 6010 students have viewed full stepbystep solutions from this chapter.

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

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

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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