 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 5386 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Outer product uv T
= column times row = rank one matrix.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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 ().

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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