 7.1.1: Find loo and I2 norms ofthe vectors. a. x = (3, 4. 0, )' b. x = (...
 7.1.2: Find Igo and I2 norms of the vectors. a. x = (2, 2, 1)' b. x = (4...
 7.1.3: Prove that the following sequences are convergent and find their li...
 7.1.4: Prove that the following sequences are convergent and find their li...
 7.1.5: Find the /ao norm ofthe matrices.
 7.1.6: Find the norm ofthe matrices. 10 I 1 11 a. b. d. b. 10 0 15 1 4 1...
 7.1.7: The following linear systems Ax = b have x as the actual solution a...
 7.1.8: The following linear systems Ax = Compute x xlloo and  Ax b a....
 7.1.9: a. Verify that the function  * II1, defined on R" by Ixlli = 51 l...
 7.1.10: The matrix norm  i, defined by /ti = max /lxi, can be c...
 7.1.11: Show by example that  oo, defined by  AHoo = max a,:;, does...
 7.1.12: Show that  , defined by 11. Show by example that  oo, defin...
 7.1.13: a. The Frobenius norm (which is not a natural norm) is defined for ...
 7.1.14: In Exercise 13, the Frobenius norm of a matrix was defined. Show th...
 7.1.15: Let S be a positive definite n x n matrix. For any x in E" define ...
 7.1.16: Let S be a real and nonsingular matrix and let   be any norm on...
 7.1.17: Prove that if  II is a vector norm on R", then  A = maXX=...
 7.1.18: The following excerpt from the Mathematics Magazine [Sz] gives an a...
 7.1.19: Show that the CauchyBuniakowskySchwarz Inequality can be strength...
Solutions for Chapter 7.1: Norms of Vectors and Matrices
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 7.1: Norms of Vectors and Matrices
Get Full SolutionsThis textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. Since 19 problems in chapter 7.1: Norms of Vectors and Matrices have been answered, more than 15052 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.1: Norms of Vectors and Matrices includes 19 full stepbystep solutions.

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].

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

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pascal matrix
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).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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