 7.4.1: Determine F, , and 1 for each of the following matrices: (a) 1 0 0 ...
 7.4.2: Let A = 2 0 0 2 and x = x1 x2 and set f(x1, x2) = Ax 2/ x 2 Determi...
 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 00 0 5 00 0 0 2 0 0 0 04 (a) Compute the singular value...
 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, and ...
 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 Rn, an...
 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 2 for...
 7.4.12: Let A = 3 1 2 1 2 7 414 (a) Determine A . (b) Find a vector x whose...
 7.4.13: Theorem 7.4.2 states that A = max 1im n j=1 aij Prove this in two...
 7.4.14: Show that A F = AT F.
 7.4.15: Let A be a symmetric n n matrix. Show that A = A 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 circu...
 7.4.18: Let denote a family of vector norms and let M be a subordinate matr...
 7.4.19: Let A be an m n matrix and let v and w be vector norms on Rn and Rm...
 7.4.20: Let A be an m n matrix. The (1,2)norm of A is given by A (1,2) = m...
 7.4.21: Let A be an mn matrix. Show that A (1,2) A 2
 7.4.22: Let A Rmn and B Rnr . Show that (a) Ax 2 A (1,2) x 1 for all x in R...
 7.4.23: Let A be an n n matrix and let M be a matrix norm that is compatibl...
 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 i...
 7.4.26: Let Aij be a submatrix of a sudoku matrix A (see Exercise 25). Show...
 7.4.27: Let A be an n n matrix and x Rn. Prove: (a) Ax n1/2 A 2 x (b) Ax 2 ...
 7.4.28: Let A be a symmetric n n matrix with eigenvalues 1, ... , n and ort...
 7.4.29: Let A = 1 0.99 1 1 Find A1 and cond(A).
 7.4.30: Solve the given two systems and compare the solutions. Are the coef...
 7.4.31: Let A = 101 223 112 Calculate cond(A) = A A1 .
 7.4.32: Let A be a nonsingular n n matrix, and let M denote a matrix norm t...
 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, deter...
 7.4.35: Given A = 3 2 1 1 and b = 5 2 If twodigit decimal floatingpoint a...
 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 1 001 1 0001 , b = 5.00 1.02 1.04 1.10 An app...
 7.4.39: Let A and B be nonsingular n n matrices. Show that cond(AB) cond(A)...
 7.4.40: Let A and B be nonsingular n n matrices. Show that cond(AB) cond(A)...
 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 UVT . Show...
 7.4.45: Let A be an mn matrix with singular value decomposition UVT . Show ...
 7.4.46: . Let A be a nonsingular nn matrix and let Q be an n n orthogonal m...
 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  9th Edition
ISBN: 9780321962218
Solutions for Chapter 7.4: Matrix Norms and Condition Numbers
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Since 47 problems in chapter 7.4: Matrix Norms and Condition Numbers have been answered, more than 11950 students have viewed full stepbystep solutions from this chapter. 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).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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.

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.

Multiplier eij.
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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.

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