- 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 re...
- 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 AT A is positive semidefi...
- 6.6.11: Let A be a symmetric nn matrix with eigenvalues 1, ... , n. Show th...
- 6.6.12: Let A be a symmetric nn matrix with eigenvalues 1, ... , n. Show th...
- 6.6.13: Let A be a symmetric positive definite n n matrix and let S be a no...
- 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 | 9th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
Column space C (A) =
space of all combinations of the columns of A.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
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 .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Solvable system Ax = b.
The right side b is in the column space of A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).