- 7.2.1: Let A = 1 1 1 2 4 1 3 1 2 Factor A into a product LU, where L is lo...
- 7.2.2: Let A be the matrix in Exercise 1. Use the LU factorization of A to...
- 7.2.3: Let A and B be n n matrices and let x Rn. (a) How many scalar addit...
- 7.2.4: Let A Rmn, B Rnr , and x, y Rn. Suppose that the product AxyTB is c...
- 7.2.5: Let Eki be the elementary matrix formed by subtracting times the it...
- 7.2.6: Let A be an n n matrix with triangular factorization LU. Show that ...
- 7.2.7: If A is a symmetric nn matrix with triangular factorization LU, the...
- 7.2.8: Write an algorithm for solving the tridiagonal system a1 b1 c1 a2 ....
- 7.2.9: Let A = LU, where L is lower triangular with 1s on the diagonal and...
- 7.2.10: Suppose that A1 and the LU factorization of A have already been det...
- 7.2.11: Let A be a 3 3 matrix, and assume that A can be transformed into a ...
Solutions for Chapter 7.2: Gaussian Elimination
Full solutions for Linear Algebra with Applications | 8th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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)·(b - Ax) = o.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).