- 6.6.1: Find the matrix associated with each of the following quadratic for...
- 6.6.2: Reorder the eigenvalues in Example 2 so that 1 = 4 and 2 = 2, and r...
- 6.6.3: In each of the following, (i) find a suitable change of coordinates...
- 6.6.4: Let 1 and 2 be the eigenvalues of A = a b b c What kind of conic se...
- 6.6.5: Let A be a symmetric 2 2 matrix and let be a nonzero scalar for whi...
- 6.6.6: Which of the matrices that follow are positive definite? Negative d...
- 6.6.7: For each of the following functions, determine whether the given st...
- 6.6.8: Show that if A is symmetric positive definite, then det(A) > 0. Giv...
- 6.6.9: Show that if A is a symmetric positive definite matrix, then A is n...
- 6.6.10: Let A be a singular n n matrix. Show that ATA is positive semidefin...
- 6.6.11: Let A be a symmetric nn matrix with eigenvalues 1, . . . , n. Show ...
- 6.6.12: Let A be a symmetric positive definite matrix. Show that the diagon...
- 6.6.13: Let A be a symmetric positive definite nn matrix and let S be a non...
- 6.6.14: Let A be a symmetric positive definite nn matrix. Show that A can b...
Solutions for Chapter 6.6: Quadratic Forms
Full solutions for Linear Algebra with Applications | 8th Edition
Tv = Av + Vo = linear transformation plus shift.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
peA) = det(A - AI) has peA) = zero matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
Outer product uv T
= column times row = rank one matrix.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
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.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.