- 6.5.1: (a) Find the Laplace transform of the solution of the initial-value...
- 6.5.2: f (t) = eat and g(t) = ebt
- 6.5.3: f (t) = cost and g(t) = u2(t)
- 6.5.4: f (t) = u2(t) and g(t) = u3(t)
- 6.5.5: Compute (3 sin t) (cos 2t) by computing the integral. [Hint: Use tr...
- 6.5.6: Show that convolution is a commutative operation. In other words, s...
- 6.5.7: Suppose the solution (t) of the initial-value problem d2 y dt2 + p ...
- 6.5.8: Verify that the solution (t) of the initial-value problem dy dt + a...
- 6.5.9: Let (t) be the solution of the initial-value problem d2 y dt2 + p d...
- 6.5.10: Suppose we know (t), the solution of the initial-value problem d2 y...
- 6.5.11: Suppose y1(t) is the solution of the initial-value problem d2 y dt2...
Solutions for Chapter 6.5: CONVOLUTIONS
Full solutions for Differential Equations 00 | 4th Edition
peA) = det(A - AI) has peA) = zero matrix.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).