- 9.6.1: In each of 1 through 4, construct a suitable Liapunov function of t...
- 9.6.2: In each of 1 through 4, construct a suitable Liapunov function of t...
- 9.6.3: In each of 1 through 4, construct a suitable Liapunov function of t...
- 9.6.4: In each of 1 through 4, construct a suitable Liapunov function of t...
- 9.6.5: Consider the system of equationsdx/dt = y xf(x, y), dy/dt = x yf(x,...
- 9.6.6: A generalization of the undamped pendulum equation isd2u/dt2 + g(u)...
- 9.6.7: By introducing suitable dimensionless variables, we can write the s...
- 9.6.8: The Linard equation ( of Section 9.3) isd2udt2 + c(u)dudt + g(u) = ...
- 9.6.9: (a) A special case of the Linard equation of isd2udt2 +dudt + g(u) ...
- 9.6.10: In 10 and 11, we will prove part of Theorem 9.3.2: If the critical ...
- 9.6.11: In 10 and 11, we will prove part of Theorem 9.3.2: If the critical ...
- 9.6.12: In this problem we prove a part of Theorem 9.3.2 related to instabi...
Solutions for Chapter 9.6: Liapunovs Second Method
Full solutions for Elementary Differential Equations and Boundary Value Problems | 10th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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].
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
A symmetric matrix with eigenvalues of both signs (+ and - ).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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 ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.