 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
ISBN: 9780136009290
Solutions for Chapter 6.6: Quadratic Forms
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 6.6: Quadratic Forms includes 14 full stepbystep solutions. Since 14 problems in chapter 6.6: Quadratic Forms have been answered, more than 8442 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
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.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.