 9.3.1: In each of 1 through 3, verify that (0, 0) is a critical point, sho...
 9.3.2: In each of 1 through 3, verify that (0, 0) is a critical point, sho...
 9.3.3: In each of 1 through 3, verify that (0, 0) is a critical point, sho...
 9.3.4: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.5: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.6: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.7: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.8: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.9: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.10: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.11: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.12: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.13: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.14: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.15: In each of 4 through 15: a. Determine all critical points of the gi...
 9.3.16: Consider the autonomous system dx dt = y, dy dt = x + 2x3. a. Show ...
 9.3.17: Consider the autonomous system dx dt = x, dy dt = 2y + x3. a. Show ...
 9.3.18: The equation of motion of an undamped pendulum is d2/dt2 + 2 sin = ...
 9.3.19: a. By solving the equation for dy/dx, show that the equation of the...
 9.3.20: The motion of a certain undamped pendulum is described by the equat...
 9.3.21: Consider again the pendulum equations (see 20) dx dt = y, dy dt = 4...
 9.3.22: This problem extends to a damped pendulum. The equations of motion ...
 9.3.23: Theorem 9.3.3 provides no information about the stability of a crit...
 9.3.24: In this problem we show how small changes in the coefficients of a ...
 9.3.25: In this problem we show how small changes in the coefficients of a ...
 9.3.26: In this problem we derive a formula for the natural period of an un...
 9.3.27: A generalization of the damped pendulum equation discussed in the t...
Solutions for Chapter 9.3: Locally Linear Systems
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 9.3: Locally Linear Systems
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 27 problems in chapter 9.3: Locally Linear Systems have been answered, more than 12363 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.3: Locally Linear Systems includes 27 full stepbystep solutions.

Cholesky factorization
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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.