 20.4.20.1.57: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.58: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.59: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.60: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.61: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.62: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.63: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.64: Compute (5), (6), (7). Compute a corresponding unit vector (vector ...
 20.4.20.1.65: Show that Ilxll oo ~ IIxl12 ~ Ilxlli.
 20.4.20.1.66: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.67: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.68: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.69: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.70: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.71: Compute the matrix norm and the condition number corresponding to t...
 20.4.20.1.72: Verify (II) for x = [4 5 2]T taken with the loenorm and the matri...
 20.4.20.1.73: Verify (12) for the matrices in Probs. to and 1 L
 20.4.20.1.74: Verify the calculations in Examples 5 and 6 of the text.
 20.4.20.1.75: Solve Ax = bb Ax = b2 compare the soLutions. and comment. Compute t...
 20.4.20.1.76: Solve Ax = bb Ax = b2 compare the soLutions. and comment. Compute t...
 20.4.20.1.77: (Residual) For Ax = b] in Prob. 19 guess what the residuaL of x = [...
 20.4.20.1.78: Show that K(A) ~ 1 for the matrix norms (10), (11), Sec. 20.3, and ...
 20.4.20.1.79: CAS EXPERIMENT. Hilbert Matrices. The 3 X 3 Hilbert matrix is The n...
 20.4.20.1.80: TEAM PROJECT. Norms. (a) Vector norms in our text are equivalent, t...
 20.4.20.1.81: WRITING PROJECT. Norms and Their Use inThis Section. Make a list of...
Solutions for Chapter 20.4: Linear Systems: IIIConditioning. Norms
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 20.4: Linear Systems: IIIConditioning. Norms
Get Full SolutionsChapter 20.4: Linear Systems: IIIConditioning. Norms includes 25 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 25 problems in chapter 20.4: Linear Systems: IIIConditioning. Norms have been answered, more than 48897 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: Advanced Engineering Mathematics, edition: 9.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

Companion matrix.
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 nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.