 8.2.1: For each of the quadratic forms q listed in Exercises 1 through 3, ...
 8.2.2: For each of the quadratic forms q listed in Exercises 1 through 3, ...
 8.2.3: For each of the quadratic forms q listed in Exercises 1 through 3, ...
 8.2.4: Determine the definiteness of the quadratic forms in Exercises 4 th...
 8.2.5: Determine the definiteness of the quadratic forms in Exercises 4 th...
 8.2.6: Determine the definiteness of the quadratic forms in Exercises 4 th...
 8.2.7: Determine the definiteness of the quadratic forms in Exercises 4 th...
 8.2.8: If A is a symmetric matrix, what can you say about the definiteness...
 8.2.9: Recall that a real square matrix A is called skew symmetric if At =...
 8.2.10: Consider a quadratic form q(x) = * Ax on R'1 and a fixed vector v i...
 8.2.11: If A is an invertible symmetric matrix, what is the relationship be...
 8.2.12: Show that a quadratic form q(x) = * Ax of two variables is indefini...
 8.2.13: Show that the diagonal elements of a positive definite matrix A are...
 8.2.14: Consider a 2 x 2 matrix A =a b b c, where a anddetA are both positi...
 8.2.15: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.16: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.17: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.18: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.19: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.20: Sketch the curves defined in Exercises 15 through 20. In each case,...
 8.2.21: a. Sketch the following three surfaces:*2 +4*2 +9*3 = 1,X2 + 4*2  ...
 8.2.22: On the surface*2 + *2 *3 + 10* 1*3 = 11 find the two points closest...
 8.2.23: Consider an n x n matrix M that is not symmetric, and define the fu...
 8.2.24: Consider a quadratic formq(x) = * Ajc,where A is a symmetric nxn ma...
 8.2.25: Consider a quadratic formq(x) = * Ax, where A is a symmetric nxn ma...
 8.2.26: Consider a quadratic formq(x) = x A*,where A is a symmetric nxn mat...
 8.2.27: Consider a quadratic form q(x) = x Ax, where A is a symmetric n x n...
 8.2.28: Show that any positive definite n x n matrix A can be written as A ...
 8.2.29: For the matrix A = write A = BBtas discussed in Exercise 28. See Ex...
 8.2.30: Show that any positive definite matrix A can be written as A = B2, ...
 8.2.31: For the matrix A =1 2 2 5 cussed in Exercise 30. See Example 1. w...
 8.2.32: Cholesky factorization for 2x2 matrices. Show that any positive def...
 8.2.33: Find the Cholesky factorization (discussed in Exercise 32) forA =8 ...
 8.2.34: A Cholesky factorization of a symmetric matrix A is a factorization...
 8.2.35: Find the Cholesky factorization of the matrix4 4 8 A = ' 4 8 13 1...
 8.2.36: Consider an invertible nxn matrix A. What is the relationship betwe...
 8.2.37: Consider the quadratic form<?(*! 1*2) = CLx\ + b x }X2 + CX2We defi...
 8.2.38: For which values of the constants p and q is the n x n matrixB =4 P...
 8.2.39: For which angles 0 can you find a basis of R such that the angle be...
 8.2.40: Show that for every symmetric n x n matrix A there exists a constan...
 8.2.41: Find the dimension of the space Q2 of all quadratic forms in two va...
 8.2.42: Find the dimension of the space Qn of all quadratic forms in n vari...
 8.2.43: Consider the transformation T{q(x\, *2)) = < 7(*i,0) from 02 to P2....
 8.2.44: Consider the transformation T{q(x\, *2)) = q(x 1, 1) from Q2 to P2....
 8.2.45: Consider the transformation T (q (x 1, *2, *3)) = q(x 1, 1, 1) from...
 8.2.46: Consider the linear transformation T[q(x\, *2, * 3)) = <?(*1, *2* *...
 8.2.47: Consider the function T(A)(x) = x T Ax from Rnxn to Q. Show that T ...
 8.2.48: Consider the linear transformation T [q(x\, *2)) = q(*2<*l) from Q2...
 8.2.49: Consider the linear transformation T (q(x\, *2)) = q(x\, 2x2) from ...
 8.2.50: Consider the linear transformationT{q(x\,x2)) = x\^~ v ' dx2 d*fro...
 8.2.51: What are the signs of the determinants of the principal submatrices...
 8.2.52: Consider a quadratic form q. If A is a symmetric matrix such that q...
 8.2.53: Consider a quadratic form q(x\,..., *) with symmetric matrix A. For...
 8.2.54: If A is a positive semidefinite matrix with a\[ =0, what can you sa...
 8.2.55: If A is a positive definite n x n matrix, show that the largest ent...
 8.2.56: If A is a real symmetric matrix, show that there exists an eigenval...
 8.2.57: In Exercises 57 through 61, consider a quadratic form q on M3 with ...
 8.2.58: In Exercises 57 through 61, consider a quadratic form q on M3 with ...
 8.2.59: In Exercises 57 through 61, consider a quadratic form q on M3 with ...
 8.2.60: In Exercises 57 through 61, consider a quadratic form q on M3 with ...
 8.2.61: In Exercises 57 through 61, consider a quadratic form q on M3 with ...
 8.2.62: Consider an indefinite quadratic form q on R3 with symmetric matrix...
 8.2.63: Consider a positive definite quadratic form q on Rn with symmetric ...
 8.2.64: For the quadratic form q(x),x2) = 4x\x2 + 5*2> find an orthogonal b...
 8.2.65: Show that for every indefinite quadratic form q on R2 there exists ...
 8.2.66: For the quadratic form q(x\,x2) = 3jc2 10* 1*2 + 3*2* find an ortho...
 8.2.67: Consider a quadratic form q on W1 with symmetric ma * trix A, with...
 8.2.68: If q is a quadratic form on Rn with symmetric matrix A, and if L(x)...
 8.2.69: If A is a positive definite nxn matrix, and R is any real n x m mat...
 8.2.70: If A is an indefinite nxn matrix, and R is a real n xm matrix of ra...
 8.2.71: If A is an indefinite nxn matrix, and R is any real nxm matrix, wha...
Solutions for Chapter 8.2: Quadratic Forms
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 8.2: Quadratic Forms
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 71 problems in chapter 8.2: Quadratic Forms have been answered, more than 15571 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.2: Quadratic Forms includes 71 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

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.

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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 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.

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).