- 7.8.1: In each of 1 through 3: a. Draw a direction field and sketch a few ...
- 7.8.2: In each of 1 through 3: a. Draw a direction field and sketch a few ...
- 7.8.3: In each of 1 through 3: a. Draw a direction field and sketch a few ...
- 7.8.4: In each of 4 and 5, find the general solution of the given system o...
- 7.8.5: In each of 4 and 5, find the general solution of the given system o...
- 7.8.6: In each of 6 through 8: a. Find the solution of the given initial v...
- 7.8.7: In each of 6 through 8: a. Find the solution of the given initial v...
- 7.8.8: In each of 6 through 8: a. Find the solution of the given initial v...
- 7.8.9: In each of 9 and 10: a. Find the solution of the given initial valu...
- 7.8.10: In each of 9 and 10: a. Find the solution of the given initial valu...
- 7.8.11: In each of 11 and 12, solve the given system of equations by the me...
- 7.8.12: In each of 11 and 12, solve the given system of equations by the me...
- 7.8.13: Show that all solutions of the system x _ = _a b c d _ x approach z...
- 7.8.14: Consider again the electric circuit in of Section 7.6. This circuit...
- 7.8.15: Consider again the system x _ = Ax = _1 1 1 3 _ x (36) that we disc...
- 7.8.16: In Example 2, with A given in equation (36) above, it was claimed t...
- 7.8.17: Consider the system x _ = Ax = 1 1 1 2 1 1 3 2 4 x. (38) a. Show th...
- 7.8.18: Consider the system x _ = Ax = 5 3 2 8 5 4 4 3 3 x. (39) a. Show th...
- 7.8.19: Let J = _ 1 0 _, where is an arbitrary real number. a. Find J2, J3,...
- 7.8.20: Let J = 0 0 0 1 0 0 , where is an arbitrary real number. a. Find J2...
- 7.8.21: Let J = 1 0 0 1 0 0 , where is an arbitrary real number. a. Find J2...
Solutions for Chapter 7.8: Repeated Eigenvalues
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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.
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].
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.