- 126.96.36.199.137: Tridiagonalize. showing the details:
- 188.8.131.52.138: Tridiagonalize. showing the details:
- 184.108.40.206.139: Tridiagonalize. showing the details:
- 220.127.116.11.140: Tridiagonalize. showing the details:
- 18.104.22.168.141: Do three QR-steps to find approximations of the eigenvalues of:The ...
- 22.214.171.124.142: Do three QR-steps to find approximations of the eigenvalues of:The ...
- 126.96.36.199.143: Do three QR-steps to find approximations of the eigenvalues of:
- 188.8.131.52.144: Do three QR-steps to find approximations of the eigenvalues of:
- 184.108.40.206.145: Do three QR-steps to find approximations of the eigenvalues of:
- 220.127.116.11.146: CAS EXPERIMENT. QR-Method. Try to find out experimentally on what p...
Solutions for Chapter 20.9: Tridiagonalization and QR-Factorization
Full solutions for Advanced Engineering Mathematics | 9th Edition
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Invert A by row operations on [A I] to reach [I A-I].
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).