- 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
peA) = det(A - AI) has peA) = zero matrix.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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].
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.