Explanation Given information: 11 x 1 2 + 11 x 2 2 + 11 x 3 2 + 11 x 4 2 + 16 x 1 x 2 − 12 x 1 x 4 + 12 x 2 x 3 + 16 x 3 x 4 If x define a variable vector in R n , then a change of variable is an equation of the form x = P y , or equivalently, y = P − 1 x Here, P is an invertible matrix and y is a new variable vector in R n . Since, y is the coordinate vector of x relative to the basis of R n determined by the columns of P If the change of variable y = P − 1 x is made in a quadratic form x T A x , then x T A x = P y T A P y = y T P T A P y = y T P T A P y and P T A P is the new matrix. Since A is symmetric, then there is an orthogonal matrix P such that P r A P is the diagonal matrix D , and the quadratic form in x T A x = y T P T A P y becomes y T D y . The Matrix of the quadratic form is A = 11 8 0 − 6 8 11 6 0 0 6 11 8 − 6 0 8 11 Now, find the eigenvalues of A det λ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 − 11 8 0 − 6 8 11 6 0 0 6 11 8 − 6 0 8 11 = 0 det λ 0 0 0 0 λ 0 0 0 0 λ 0 0 0 0 λ − 11 8 0 − 6 8 11 6 0 0 6 11 8 − 6 0 8 11 = 0 det λ − 11 − 8 0 6 − 8 λ − 11 − 6 0 0 − 6 λ − 11 − 8 6 0 − 8 λ − 11 = 0 By solving the above determinant, the eigenvalues of A are λ 1 = 21 , λ 2 = 1 So, the quadratic form is indefinite because it assumes both positive and negative values

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Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 7.2, Problem 17E

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To classify: The given quadratic form. Then make a change of variable, x=Py , that transforms the quadratic form into one with no cross-product term.

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Chapter 7.2, Problem 16E

Chapter 7.2, Problem 18E

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Convert the quadratic form to canonical form.

Chapter 7 Solutions

EBK LINEAR ALGEBRA AND ITS APPLICATIONS

Ch. 7.1 - Determine which of the matrices in Exercises 7-12...Ch. 7.1 - Determine which of the matrices in Exercises 7-12...Ch. 7.1 - Determine which of the matrices in Exercises 7-12...Ch. 7.1 - Determine which of the matrices in Exercises 7-12...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Orthogonally diagonalize the matrices in Exercises...Ch. 7.1 - Prob. 22E Ch. 7.1 - Let A=[411141114]andv=[111]. Verify that 5 is an...Ch. 7.1 - Let A=[211121112],v1=[101],andv2=[111]. Verify...Ch. 7.1 - Prob. 25E Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - Prob. 30E Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - In Exercises 25â€”32, mark each statement True or...Ch. 7.1 - Show that if A is an n n symmetric matrix, then...Ch. 7.1 - Suppose A is a symmetric n n matrix and B is any...Ch. 7.1 - Suppose A is invertible and orthogonally...Ch. 7.1 - Suppose A and B are both orthogonally...Ch. 7.1 - Let A = PDP1, where P is orthogonal and D is...Ch. 7.1 - Suppose A = PRP1, where P is orthogonal and R is...Ch. 7.1 - Construct a spectral decomposition of A from...Ch. 7.1 - Construct a spectral decomposition of A from...Ch. 7.1 - Prob. 41E Ch. 7.1 - Let B be an n n symmetric matrix such that B2 =...Ch. 7.1 - Prob. 43E Ch. 7.2 - Describe a positive semidefinite matrix A in terms...Ch. 7.2 - Compute the quadratic form XTAX, when A=[51/31/31]...Ch. 7.2 - Prob. 2E Ch. 7.2 - Find the matrix of the quadratic form. Assume x is...Ch. 7.2 - Find the matrix of the quadratic form. Assume x is...Ch. 7.2 - Find the matrix of the quadratic form. Assume x is...Ch. 7.2 - Find the matrix of the quadratic form. Assume x is...Ch. 7.2 - Make a change of variable, x = Py, that transforms...Ch. 7.2 - Let A be the matrix of the quadratic form...Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Classify the quadratic forms in Exercises 9-18....Ch. 7.2 - Prob. 17E Ch. 7.2 - What is the largest possible value of the...Ch. 7.2 - What is the largest value of the quadratic form...Ch. 7.2 - Prob. 21E Ch. 7.2 - Prob. 22E Ch. 7.2 - Prob. 23E Ch. 7.2 - Prob. 24E Ch. 7.2 - Prob. 25E Ch. 7.2 - Prob. 26E Ch. 7.2 - Prob. 27E Ch. 7.2 - Prob. 28E Ch. 7.2 - Prob. 29E Ch. 7.2 - Prob. 30E Ch. 7.2 - Exercises 23 and 24 show how to classify a...Ch. 7.2 - Exercises 23 and 24 show how to classify a...Ch. 7.2 - Show that if B is m n, then BTB is positive...Ch. 7.2 - Prob. 34E Ch. 7.2 - Let A and B be symmetric n n matrices whose...Ch. 7.2 - Let A be an n n invertible symmetric matrix. Show...Ch. 7.3 - Let Q(x)=3x12+3x22+2x1x2. Find a change of...Ch. 7.3 - Prob. 2PP Ch. 7.3 - In Exercises 1 and 2, find the change of variable...Ch. 7.3 - In Exercises 1 and 2, find the change of variable...Ch. 7.3 - In Exercises 3-6, find (a) the maximum value of...Ch. 7.3 - In Exercises 3-6, find (a) the maximum value of...Ch. 7.3 - In Exercises 3-6, find (a) the maximum value of...Ch. 7.3 - In Exercises 3-6, find (a) the maximum value of...Ch. 7.3 - Let Q(x)=2x12x22+4x1x2+4x2x3. Find a unit vector x...Ch. 7.3 - Let Q(x)=7x12+x22+7x324x1x24x1x3. Find a unit...Ch. 7.3 - Find the maximum value of Q(x)=7x12+3x222x1x2,...Ch. 7.3 - Find the maximum value of Q(x)=3x12+5x222x1x2,...Ch. 7.3 - Suppose x is a unit eigenvector of a matrix A...Ch. 7.3 - Prob. 12E Ch. 7.3 - Prob. 13E Ch. 7.3 - Prob. 14E Ch. 7.3 - Prob. 15E Ch. 7.3 - Prob. 16E Ch. 7.3 - In Exercises 3-6, find (a) the maximum value of...Ch. 7.4 - Given a singular value decomposition, A = UVT,...Ch. 7.4 - Prob. 2PP Ch. 7.4 - Find the singular values of the matrices in...Ch. 7.4 - Find the singular values of the matrices in...Ch. 7.4 - Find the singular values of the matrices in...Ch. 7.4 - Find the singular values of the matrices in...Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find an SVD of each matrix in Exercises 512....Ch. 7.4 - Find the SVD of A=[322232] [Hint: Work with AT.]Ch. 7.4 - In Exercise 7, find a unit vector x at which Ax...Ch. 7.4 - Suppose the factorization below is an SVD of a...Ch. 7.4 - Prob. 16E Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - Prob. 21E Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - Prob. 23E Ch. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - Prob. 25E Ch. 7.4 - Prob. 28E Ch. 7.4 - Prob. 29E Ch. 7.5 - The following table lists the weights and heights...Ch. 7.5 - The following table lists the weights and heights...Ch. 7.5 - In Exercises 1 and 2, convert the matrix of...Ch. 7.5 - In Exercises 1 and 2, convert the matrix of...Ch. 7.5 - Find the principal components of toe data for...Ch. 7.5 - Find the principal components of the data for...Ch. 7.5 - [M] A Landsat image with three spectral components...Ch. 7.5 - [M] The covariance matrix below was obtained from...Ch. 7.5 - Prob. 7E Ch. 7.5 - Prob. 8E Ch. 7.5 - Suppose three tests are administered to a random...Ch. 7.5 - [M] Repeal Exercise 9 with S=[5424114245]. 9....Ch. 7.5 - Prob. 11E Ch. 7.5 - Prob. 12E Ch. 7.5 - The sample covariance matrix is a generalization...Ch. 7 - Prob. 1SE Ch. 7 - Prob. 2SE Ch. 7 - Prob. 3SE Ch. 7 - Prob. 4SE Ch. 7 - Mark each statement True or False. Justify each...Ch. 7 - Prob. 6SE Ch. 7 - Prob. 7SE Ch. 7 - Prob. 8SE Ch. 7 - Prob. 9SE Ch. 7 - Prob. 10SE Ch. 7 - Prob. 11SE Ch. 7 - Prob. 12SE Ch. 7 - Prob. 13SE Ch. 7 - Prob. 14SE Ch. 7 - Prob. 15SE Ch. 7 - Prob. 16SE Ch. 7 - Prob. 17SE Ch. 7 - Prob. 18SE Ch. 7 - Let A be an n n symmetric matrix of rank r....Ch. 7 - Let A be an n n symmetric matrix. a. Show that...Ch. 7 - Prob. 21SE Ch. 7 - Prob. 22SE Ch. 7 - Prob. 23SE Ch. 7 - Prob. 24SE Ch. 7 - If A is m n, then the matrix G = ATA is called...Ch. 7 - If A is m n, then the matrix G = ATA is called...Ch. 7 - Prove that any n n matrix A admits a polar...Ch. 7 - Prob. 28SE Ch. 7 - Prob. 30SE

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To classify: The given quadratic form. Then make a change of variable, x = P y , that transforms the quadratic form into one with no cross-product term.

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To classify: The given quadratic form. Then make a change of variable, x=Py , that transforms the quadratic form into one with no cross-product term.

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Chapter 7 Solutions