 2.7.1: Write all 33 elementary matrices and their inverses.
 2.7.2: For 26, determine elementary matrices that reduce the given matrix ...
 2.7.3: For 26, determine elementary matrices that reduce the given matrix ...
 2.7.4: For 26, determine elementary matrices that reduce the given matrix ...
 2.7.5: For 26, determine elementary matrices that reduce the given matrix ...
 2.7.6: For 26, determine elementary matrices that reduce the given matrix ...
 2.7.7: For 713, express the matrix A as a product of elementary matrices.
 2.7.8: For 713, express the matrix A as a product of elementary matrices.
 2.7.9: For 713, express the matrix A as a product of elementary matrices.
 2.7.10: For 713, express the matrix A as a product of elementary matrices.
 2.7.11: For 713, express the matrix A as a product of elementary matrices.
 2.7.12: For 713, express the matrix A as a product of elementary matrices.
 2.7.13: For 713, express the matrix A as a product of elementary matrices.
 2.7.14: Determine elementary matrices E1, E2,..., Ek that reduce A = 2 1 1 ...
 2.7.15: Determine a Type 3 lower triangular elementary matrix E1 that reduc...
 2.7.16: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.17: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.18: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.19: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.20: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.21: For 1621, determine the LU factorization of the given matrix. Verif...
 2.7.22: For 2225, use the LU factorization of A to solve the system Ax = b....
 2.7.23: For 2225, use the LU factorization of A to solve the system Ax = b.
 2.7.24: For 2225, use the LU factorization of A to solve the system Ax = b....
 2.7.25: For 2225, use the LU factorization of A to solve the system Ax = b....
 2.7.26: Use the LU factorization of A = 2 1 8 3 to solve each of the system...
 2.7.27: Use the LU factorization of A = 1 42 3 14 5 7 1 to solve each of th...
 2.7.28: If P = P1P2 ... Pk , where each Pi is an elementary permutation mat...
 2.7.29: Prove that (a) the inverse of an invertible upper triangular matrix...
 2.7.30: In this problem, we prove that the LU decomposition of an invertibl...
 2.7.31: QR Factorization: It can be shown that any invertible n n matrix ha...
 2.7.32: For 3234, use some form of technology to determine the LU factoriza...
 2.7.33: For 3234, use some form of technology to determine the LU factoriza...
 2.7.34: For 3234, use some form of technology to determine the LU factoriza...
Solutions for Chapter 2.7: Elementary Matrices and the LU Factorization
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 2.7: Elementary Matrices and the LU Factorization
Get Full SolutionsChapter 2.7: Elementary Matrices and the LU Factorization includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Since 34 problems in chapter 2.7: Elementary Matrices and the LU Factorization have been answered, more than 19262 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·