 7.6.1: In each of 1 through 4: G a. Draw a direction field and sketch a fe...
 7.6.2: In each of 1 through 4: G a. Draw a direction field and sketch a fe...
 7.6.3: In each of 1 through 4: G a. Draw a direction field and sketch a fe...
 7.6.4: In each of 1 through 4: G a. Draw a direction field and sketch a fe...
 7.6.5: In each of 5 and 6, express the general solution of the given syste...
 7.6.6: In each of 5 and 6, express the general solution of the given syste...
 7.6.7: In each of 7 and 8, find the solution of the given initialvalue pr...
 7.6.8: In each of 7 and 8, find the solution of the given initialvalue pr...
 7.6.9: In each of 9 and 10: a. Find the eigenvalues of the given system. G...
 7.6.10: In each of 9 and 10: a. Find the eigenvalues of the given system. G...
 7.6.11: In each of 11 through 15, the coefficient matrix contains a paramet...
 7.6.12: In each of 11 through 15, the coefficient matrix contains a paramet...
 7.6.13: In each of 11 through 15, the coefficient matrix contains a paramet...
 7.6.14: In each of 11 through 15, the coefficient matrix contains a paramet...
 7.6.15: In each of 11 through 15, the coefficient matrix contains a paramet...
 7.6.16: In each of 16 and 17, solve the given system of equations by the me...
 7.6.17: In each of 16 and 17, solve the given system of equations by the me...
 7.6.18: In each of 18 and 19: a. Find the eigenvalues of the given system. ...
 7.6.19: In each of 18 and 19: a. Find the eigenvalues of the given system. ...
 7.6.20: Consider the electric circuit shown in Figure 7.6.6. Suppose that R...
 7.6.21: The electric circuit shown in Figure 7.6.7 is described by the syst...
 7.6.22: In this problem we indicate how to show that u(t) and v(t), as give...
 7.6.23: A mass m on a spring with constant k satisfies the differential equ...
 7.6.24: Consider the twomass, threespring system of Example 3 in the text...
 7.6.25: Consider the twomass, threespring system whose equations of motio...
Solutions for Chapter 7.6: ComplexValued Eigenvalues
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 7.6: ComplexValued Eigenvalues
Get Full SolutionsChapter 7.6: ComplexValued Eigenvalues includes 25 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Since 25 problems in chapter 7.6: ComplexValued Eigenvalues have been answered, more than 13244 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt 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.

Graph G.
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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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).

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).