 7.6.1: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.2: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.3: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.4: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.5: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.6: In each of 1 through 6: (a) Express the general solution of the giv...
 7.6.7: In each of 7 and 8 express the general solution of the given system...
 7.6.8: In each of 7 and 8 express the general solution of the given system...
 7.6.9: In each of 9 and 10 find the solution of the given initial value pr...
 7.6.10: In each of 9 and 10 find the solution of the given initial value pr...
 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 paramete...
 7.6.14: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.15: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.16: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.17: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.18: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.19: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.20: In each of 13 through 20 the coefficient matrix contains a paramete...
 7.6.21: In each of 21 and 22 solve the given system of equations by the met...
 7.6.22: In each of 21 and 22 solve the given system of equations by the met...
 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  9th Edition
ISBN: 9780470383346
Solutions for Chapter 7.6: Complex Eigenvalues
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Since 31 problems in chapter 7.6: Complex Eigenvalues have been answered, more than 14037 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: Complex Eigenvalues includes 31 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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.

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].

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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