- LAB 4.2.1: Assuming R, C, and L are all nonnegative, what types of long-term b...
- LAB 4.2.2: In a typical circuit R is on the order of 1000, C is on the order o...
- LAB 4.2.3: Describe the solutions for various values of a and if R = 2000, C =...
Solutions for Chapter LAB 4.2: A Periodically Forced RLC Circuit
Full solutions for Differential Equations 00 | 4th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Upper triangular systems are solved in reverse order Xn to Xl.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.