 7.3.1: Let A = 0 3 1 1 2 2 2 5 4 and b = 1 7 1 (a) Reorder the rows of (A...
 7.3.2: Let A be the matrix in Exercise 1. Use the factorization PTLU to so...
 7.3.3: Let A = 1 8 6 1 4 5 2 4 6 and b = 8 1 4 Solve the system Ax = b usi...
 7.3.4: Let A = 3 2 2 4 and b = 5 2 Solve the system Ax = b using complete ...
 7.3.5: Let A be the matrix in Exercise 4 and let c = (6,4)T . Solve the sy...
 7.3.6: Let A = 5 4 7 2 4 3 2 8 6 , b = 2 5 4 , c = 5 4 2 (a) Use complete ...
 7.3.7: The exact solution of the system 0.6000x1 + 2000x2 = 2003 0.3076x1 ...
 7.3.8: Solve the system in Exercise 7 using fourdigit decimal floatingpo...
 7.3.9: Solve the system in Exercise 7 using fourdigit decimal floatingpo...
 7.3.10: Use fourdigit decimal floatingpoint arithmetic, and scale the sys...
Solutions for Chapter 7.3: Pivoting Strategies
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7.3: Pivoting Strategies
Get Full SolutionsLinear 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. Since 10 problems in chapter 7.3: Pivoting Strategies have been answered, more than 4348 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.3: Pivoting Strategies includes 10 full stepbystep solutions.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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).

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

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

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

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.

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

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