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# 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

Solutions for Chapter 7.6
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##### ISBN: 9780470383346

Elementary 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 step-by-step solutions from this chapter. Chapter 7.6: Complex Eigenvalues includes 31 full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• 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 Fn-1c 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 e-iO , 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).

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