 7.4.1: Determine _ _F , _ _, and _ _1 for each of the following matrices: ...
 7.4.2: Let A = 2 0 0 2 and x = x1 x2 and set f (x1, x2) = _Ax_2/_x_2 Deter...
 7.4.3: Let A = 1 0 0 0 Use the method of Exercise 2 to determine the value...
 7.4.4: Let D = 3 0 0 0 0 5 0 0 0 0 2 0 0 0 0 4 (a) Compute the singular va...
 7.4.5: Show that if D is an n n diagonal matrix, then _D_2 = max 1in (dii )
 7.4.6: If D is an n n diagonal matrix, how do the values of _D_1, _D_2, an...
 7.4.7: Let I denote the n n identity matrix. Determine the values of _I _1...
 7.4.8: Let _ _M denote a matrix norm on Rnn, _ _V denote a vector norm on ...
 7.4.9: A vector x in Rn can also be viewed as an n 1 matrix X: x = X = x1 ...
 7.4.10: A vector y in Rn can also be viewed as an n 1 matrix Y = (y). Show ...
 7.4.11: Let A = wyT , where w Rm and y Rn. Show that (a) _Ax_2 _x_2 _y_2_w_...
 7.4.12: Let A = 3 1 2 1 2 7 4 1 4 (a) Determine _A_. (b) Find a vector x wh...
 7.4.13: Theorem 7.4.2 states that _A_ = max 1im _n j=1 ai j  Prove this i...
 7.4.14: Show that _A_F = _AT _F . 1
 7.4.15: Let A be a symmetric n n matrix. Show that _A_ = _A_1. 1
 7.4.16: Let A be a 5 4 matrix with singular values 1 = 5, 2 = 3, and 3 = 4 ...
 7.4.17: Let A be an m n matrix. (a) Show that _A_2 _A_F . (b) Under what ci...
 7.4.18: Let _ _ denote the family of vector norms and let _ _M be a subordi...
 7.4.19: Let A be an m n matrix and let _ _v and _ _w be vector norms on Rn ...
 7.4.20: Let A be an m n matrix. The 1,2norm of A is given by _A_1,2 = max ...
 7.4.21: Let A be an m n matrix. Show that _A_1,2 _A_2 2
 7.4.22: Let A be an m n matrix and let B Rnr . Show that (a) _Ax_ _A_1,2_x_...
 7.4.23: Let A be an n n matrix and let _ _M be a matrix norm that is compat...
 7.4.24: Use the result from Exercise 23 to show that if is an eigenvalue of...
 7.4.25: Sudoku is a popular puzzle involving matrices. In this puzzle, one ...
 7.4.26: Let Ai j be a submatrix of a sudoku matrix A (see Exercise 25). Sho...
 7.4.27: Let A be an n n matrix and x Rn. Prove: (a) _Ax_ n1/2_A_2_x_ (b) _A...
 7.4.28: Let A be a symmetric n n matrix with eigenvalues 1, . . . , n and o...
 7.4.29: Let A = 1 0.99 1 1 Find A1 and cond(A). 3
 7.4.30: Solve the given two systems and compare the solutions. Are the coef...
 7.4.31: Let A = 1 0 1 2 2 3 1 1 2 Calculate cond(A) = _A__A1_. 3
 7.4.32: Let A be a nonsingular n n matrix, and let _ _M denote a matrix nor...
 7.4.33: Let An = 1 1 1 1 1 n for each positive integer n. Calculate (a) A1 ...
 7.4.34: If A is a 53 matrix with _A_2 = 8, cond2(A) = 2, and _A_F = 12, det...
 7.4.35: Let A = 3 2 1 1 and b = 5 2 The solution computed using twodigit d...
 7.4.36: Let A = 0.50 0.75 0.25 0.50 0.25 0.25 1.00 0.50 0.50 Calculate cond...
 7.4.37: Let A be the matrix in Exercise 36 and let A_ = 0.5 0.8 0.3 0.5 0.3...
 7.4.38: Let A = 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 , b = 5.00 1.02 1.04 1.10 A...
 7.4.39: Let A and B be nonsingular n n matrices. Show that cond(AB) cond(A)...
 7.4.40: Let D be a nonsingular n n diagonal matrix and let dmax = max 1in ...
 7.4.41: Let Q be an n n orthogonal matrix. Show that (a) _Q_2 = 1 (b) cond2...
 7.4.42: Let A be an n n matrix and let Q and V be n n orthogonal matrices. ...
 7.4.43: Let A be an m n matrix and let 1 be the largest singular value of A...
 7.4.44: Let A be an m n matrix with singular value decomposition U_V T . Sh...
 7.4.45: Let A be an m n matrix with singular value decomposition U_V T . Sh...
 7.4.46: Let A be a nonsingular n n matrix and let Q be an n n orthogonal ma...
 7.4.47: Let A be a symmetric nonsingular nn matrix with eigenvalues 1, . . ...
Solutions for Chapter 7.4: Matrix Norms and Condition Numbers
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7.4: Matrix Norms and Condition Numbers
Get Full SolutionsSince 47 problems in chapter 7.4: Matrix Norms and Condition Numbers have been answered, more than 4186 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 7.4: Matrix Norms and Condition Numbers includes 47 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.)

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).