Explanation An eigenvector of an n × n matrix A is a nonzero vector x such that A x = λ x for some nonzero scalar λ . A scalar λ s called eigenvalue of A if there is a nontrivial solution x of A x = λ x ; such that x is called an eigenvector corresponding to λ . An n × n matrix A is said to be orthogonally diagonalizable if there are an orthogonal matrix P and a diagonal matrix D such that A = P D P − 1 . The characteristic equation is, det ( A − λ I ) = 0 (1) Theorem used: If A is symmetric, then any two eigenvectors from different eigen spaces are orthogonal. Calculations: The eigen values of the matrix are 3 and 5. Use the below relation to calculate the basis for eigen spaces. ( A − λ I ) x = 0 (2) For λ = 3 , Substitute [ 4 0 1 0 0 4 0 1 1 0 4 0 0 1 0 4 ] for A, [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] for I and 3 for λ in Equation (2). ( A − 3 I ) x = 0 ( [ 4 0 1 0 0 4 0 1 1 0 4 0 0 1 0 4 ] − 3 [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ) x = [ 0 0 0 0 ] [ 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ] x = [ 0 0 0 0 ] Thus, the augmented matrix is, [ 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 ] . Consider row 1 and row 2, row 3 and row 4 are denoted by R 1 , R 2 , R 3 and R 4 . Now, obtain the solution of the equation ( A − I ) x = 0 using augmented matrix [ 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 ] as follows. [ 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 ] ~ [ 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 ] R 3 → − R 1 +R 3 ~ [ 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 ] R 4 → − R 2 +R 4 The solution of augmented matrix is, x 1 = − x 3 x 2 = − x 4 } Consider the value of eigen vector x is, [ x 1 x 2 x 3 x 4 ] . Then, the general solution is [ x 1 x 2 x 3 x 4 ] = x 3 [ − 1 0 1 0 ] + x 4 [ 0 − 1 0 1 ] Therefore, the basis for the eigenspace corresponding to the eigenvector λ = 3 is { [ − 1 0 1 0 ] , [ 0 − 1 0 1 ] } . Similarly, obtain the eigenvector corresponding to the eigenvalue λ = 5 . Obtain the matrix ( A − 5 I ) x = 0 as follows. ( A − 5 I ) x = 0 ( [ 4 0 1 0 0 4 0 1 1 0 4 0 0 1 0 4 ] − 5 [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ) x = [ 0 0 0 0 ] [ − 1 0 1 0 0 − 1 0 1 1 0 − 1 0 0 1 0 − 1 ] x = [ 0 0 0 0 ] Now, obtain the solution of the equation ( A − 5 I ) x = 0 using augmented matrix [ − 1 0 1 0 0 0 − 1 0 1 0 1 0 − 1 0 0 0 1 0 − 1 0 ] . Consider row 1 and row 2, row 3 and row 4 are denoted by R 1 , R 2 , R 3 and R 4

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

5th Edition

ISBN: 9781323132098

Author: Thomas, Lay

Publisher: PEARSON C

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Chapter 7.1, Problem 22E

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To diagonalize: The matrix [4010040110400104] by orthogonal diagonalization.

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Chapter 7.1, Problem 21E

Chapter 7.1, Problem 23E

Chapter 7 Solutions

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

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 - a. An n n matrix that is orthogonally...Ch. 7.1 - a. There are symmetric matrices that are not...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. 35E Ch. 7.1 - Let B be an n n symmetric matrix such that B2 =...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 - Compute the quadratic form XTAX, when...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 - What is the largest possible value of the...Ch. 7.2 - What is the largest value of the quadratic form...Ch. 7.2 - In Exercises 21 and 22, matrices are n n and...Ch. 7.2 - In Exercises 21 and 22, matrices are n n and...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. 26E 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. 7.5 - 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 - Mark each statement True or False. Justify each...Ch. 7 - Prob. 2SE 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. 5SE Ch. 7 - Let A be an n n symmetric matrix. Use Exercise 5...Ch. 7 - Prove that an n n matrix A is positive definite...Ch. 7 - Use Exercise 7 to show that if A is positive...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. 12SE Ch. 7 - Prob. 13SE Ch. 7 - Given any b in m, adapt Exercise 13 to show that...

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To diagonalize: The matrix [ 4 0 1 0 0 4 0 1 1 0 4 0 0 1 0 4 ] by orthogonal diagonalization.

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Chapter 7 Solutions