 6.5.1: (a) Find the Laplace transform of the solution of the initialvalue...
 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 initialvalue problem d2 y dt2 + p ...
 6.5.8: Verify that the solution (t) of the initialvalue problem dy dt + a...
 6.5.9: Let (t) be the solution of the initialvalue problem d2 y dt2 + p d...
 6.5.10: Suppose we know (t), the solution of the initialvalue problem d2 y...
 6.5.11: Suppose y1(t) is the solution of the initialvalue problem d2 y dt2...
Solutions for Chapter 6.5: CONVOLUTIONS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 6.5: CONVOLUTIONS
Get Full SolutionsChapter 6.5: CONVOLUTIONS includes 11 full stepbystep solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 11 problems in chapter 6.5: CONVOLUTIONS have been answered, more than 15707 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4.

CayleyHamilton Theorem.
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.

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.

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 IIA1 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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
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(DÂ» 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 AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).