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# Solutions for Chapter 9.2: Autonomous Systems and Stability

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition

ISBN: 9781119256007

Solutions for Chapter 9.2: Autonomous Systems and Stability

Solutions for Chapter 9.2
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##### ISBN: 9781119256007

This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 24 problems in chapter 9.2: Autonomous Systems and Stability have been answered, more than 12406 students have viewed full step-by-step solutions from this chapter. Chapter 9.2: Autonomous Systems and Stability includes 24 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• 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 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.

• Gram-Schmidt 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.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Linear combination cv + d w or L C jV j.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Partial pivoting.

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

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Pseudoinverse A+ (Moore-Penrose 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).

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• 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.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B II·

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