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Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
Publisher: PEARSON
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Textbook Question
Chapter 7.2, Problem 25E
Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then BTB is positive definite.
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Students have asked these similar questions
If A is invertible, which properties of A remain true for A-¹?
(a) A is triangular. (b) A is symmetric. (c) A is tridiagonal. (d) All entries are
whole numbers. (e) All entries are fractions (including numbers like 3).
Let a and b be real numbers satisfying a² - a -1 = 0 and 6² - 36 - 2 = 0,
and let
V =
1
a
b
ab
(a) Find a 4×4 matrix A with integer entries such that Av = av, and find
a 4 × 4 matrix B with integer entries such that Bv = bv. Hint: Note
that a² = 1+a and b² = 2+3b. Do not try to use the quadratic formula
to express a and b in terms of square roots, because that formula will
not help here.
(b) Show that (A+B)v = (a+b)v and that ABv = bav = abv. Take care
with the order of multiplication when treating ABv.
(c) Deduce from part (b) that there are polynomials
f(x) = x² + 3x³ + ₂x² + ₁x + co
g(x) = x² + d3x³ + d₂x² + d₁x + do
such that f(a + b) = 0 and g(ab) = 0, where Co,..., C3, do, ..., dz
are integers. You do not need to find f(x) and g(x) explicitly. Hint:
Consider characteristic polynomials.
Show that if Ad-bc=0, then the equation Ax=0 has more than one solution. Why does this imply that A is not invertible?
Chapter 7 Solutions
Linear Algebra and Its Applications (5th Edition)
Ch. 7.1 - Show that if A is a symmetric matrix, then A2 is...Ch. 7.1 - Show that if A is orthogonally diagonalizable,...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...Ch. 7.1 - Determine which of the matrices in Exercises 1-6...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 - 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. 22ECh. 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. 35ECh. 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. 26ECh. 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. 2PPCh. 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. 12ECh. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - Prob. 15ECh. 7.3 - Prob. 16ECh. 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. 2PPCh. 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. 16ECh. 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. 21ECh. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - Prob. 23ECh. 7.4 - In Exercises 1724, A is an m n matrix with a...Ch. 7.4 - Prob. 25ECh. 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. 7ECh. 7.5 - Prob. 8ECh. 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. 11ECh. 7.5 - Prob. 12ECh. 7.5 - The sample covariance matrix is a generalization...Ch. 7 - Mark each statement True or False. Justify each...Ch. 7 - Prob. 2SECh. 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. 5SECh. 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. 12SECh. 7 - Prob. 13SECh. 7 - Given any b in m, adapt Exercise 13 to show that...
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