- 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 four-digit decimal floating-po...
- 7.3.9: Solve the system in Exercise 7 using four-digit decimal floating-po...
- 7.3.10: Use four-digit decimal floating-point arithmetic, and scale the sys...
Solutions for Chapter 7.3: Pivoting Strategies
Full solutions for Linear Algebra with Applications | 8th Edition
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
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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).
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 = Q-1 = Q.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.