- 2.9.1: In each of 1 through 6, solve the given difference equation in term...
- 2.9.2: In each of 1 through 6, solve the given difference equation in term...
- 2.9.3: In each of 1 through 6, solve the given difference equation in term...
- 2.9.4: In each of 1 through 6, solve the given difference equation in term...
- 2.9.5: In each of 1 through 6, solve the given difference equation in term...
- 2.9.6: In each of 1 through 6, solve the given difference equation in term...
- 2.9.7: Find the effective annual yield of a bank account that pays interes...
- 2.9.8: An investor deposits $1000 in an account paying interest at a rate ...
- 2.9.9: A certain college graduate borrows $8000 to buy a car. The lender c...
- 2.9.10: A homebuyer wishes to take out a mortgage of $100,000 for a 30-year...
- 2.9.11: A homebuyer takes out a mortgage of $100,000 with an interest rate ...
- 2.9.12: If the interest rate on a 20-year mortgage is fixed at 10% and if a...
- 2.9.13: A homebuyer wishes to finance the purchase with a $95,000 mortgage ...
- 2.9.14: The Logistic Difference Equation. 14 through 19 deal with the diffe...
- 2.9.15: The Logistic Difference Equation. 14 through 19 deal with the diffe...
- 2.9.16: The Logistic Difference Equation. 14 through 19 deal with the diffe...
- 2.9.17: The Logistic Difference Equation. 14 through 19 deal with the diffe...
- 2.9.18: The Logistic Difference Equation. 14 through 19 deal with the diffe...
- 2.9.19: The Logistic Difference Equation. 14 through 19 deal with the diffe...
Solutions for Chapter 2.9: First Order Difference Equations
Full solutions for Elementary Differential Equations | 10th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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).
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.
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
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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).
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.
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).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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.