 7.6.1: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.2: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.3: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.4: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.5: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.6: In each of 1 through 6:(a) Express the general solution of the give...
 7.6.7: In each of 7 and 8, express the general solution of the given syste...
 7.6.8: In each of 7 and 8, express the general solution of the given syste...
 7.6.9: In each of 9 and 10, find the solution of the given initial value p...
 7.6.10: In each of 9 and 10, find the solution of the given initial value p...
 7.6.11: In each of 11 and 12:(a) Find the eigenvalues of the given system.(...
 7.6.12: In each of 11 and 12:(a) Find the eigenvalues of the given system.(...
 7.6.13: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.14: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.15: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.16: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.17: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.18: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.19: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.20: In each of 13 through 20, the coefficient matrix contains a paramet...
 7.6.21: In each of 21 and 22, solve the given system of equations by the me...
 7.6.22: In each of 21 and 22, solve the given system of equations by the me...
 7.6.23: In each of 23 and 24:(a) Find the eigenvalues of the given system.(...
 7.6.24: In each of 23 and 24:(a) Find the eigenvalues of the given system.(...
 7.6.25: Consider the electric circuit shown in Figure 7.6.6. Suppose that R...
 7.6.26: The electric circuit shown in Figure 7.6.7 is described by the syst...
 7.6.27: In this problem we indicate how to show that u(t) and v(t), as give...
 7.6.28: A mass m on a spring with constant k satisfies the differential equ...
 7.6.29: Consider the twomass, threespring system of Example 3 in the text...
 7.6.30: Consider the twomass, threespring system whose equations of motio...
 7.6.31: Consider the twomass, threespring system whose equations of motio...
Solutions for Chapter 7.6: Complex Eigenvalues
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 7.6: Complex Eigenvalues
Get Full SolutionsSince 31 problems in chapter 7.6: Complex Eigenvalues have been answered, more than 16933 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: Complex Eigenvalues includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Factorization
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.

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 columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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).

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).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).