 9.1.1: Use the method of Example 1 and the LUdecomposition 3 6 2 5 = 3 0 ...
 9.1.2: Use the method of Example 1 and the LUdecomposition 3 6 3 206 474 ...
 9.1.3: In Exercises 36, find an LUdecomposition of the coefficient matrix...
 9.1.4: In Exercises 36, find an LUdecomposition of the coefficient matrix...
 9.1.5: In Exercises 36, find an LUdecomposition of the coefficient matrix...
 9.1.6: In Exercises 36, find an LUdecomposition of the coefficient matrix...
 9.1.7: In Exercises 78, an LUdecomposition of a matrix A is given. (a) Co...
 9.1.8: In Exercises 78, an LUdecomposition of a matrix A is given. (a) Co...
 9.1.9: Let A = 2 1 1 2 1 2 210 (a) Find an LUdecomposition of A. (b) Expr...
 9.1.10: (a) Show that the matrix 0 1 1 0 has no LUdecomposition. (b) Find ...
 9.1.11: In Exercises 1112, use the given PLUdecomposition of A to solve th...
 9.1.12: In Exercises 1112, use the given PLUdecomposition of A to solve th...
 9.1.13: In Exercises 1314, find the LDUdecomposition of A. A = 2 2 4 1
 9.1.14: In Exercises 1314, find the LDUdecomposition of A. A = 3 12 6 020 ...
 9.1.15: In Exercises 1516, find a PLUdecomposition of A, and use it to sol...
 9.1.16: In Exercises 1516, find a PLUdecomposition of A, and use it to sol...
 9.1.17: Let Ax = b be a linear system of n equations in n unknowns, and ass...
 9.1.18: Let A = a b c d (a) Prove: If a = 0, then the matrix A has a unique...
 9.1.19: Prove: If A is any n n matrix, then A can be factored as A = PLU, w...
 9.1.TF: TF. In parts (a)(e) determine whether the statement is true or fals...
 9.1.T1: Technology utilities vary in how they handle LUdecompositions. For...
 9.1.T2: The accompanying figure shows a metal plate whose edges are held at...
Solutions for Chapter 9.1: LUDecompositions
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 9.1: LUDecompositions
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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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 •

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!