Linear Algebra and Its Applications (5th Edition)
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
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Chapter 7, Problem 1SE

Mark each statement True or False. Justify each answer. In each part, A represents an n × n matrix.

  1. a. If A is orthogonally diagonalizable, then A is symmetric.
  2. b. If A is an orthogonal matrix, then A is symmetric.
  3. c. If A is an orthogonal matrix, then ||Ax|| = ||x|| for all x in ℝn.
  4. d. The principal axes of a quadratic form xTAx can be the columns of any matrix P that diagonalizes A.
  5. e. If P is an n × n matrix with orthogonal columns, then PT = P−l.
  6. f. If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
  7. g. If xTAx > 0 for some x, then the quadratic form xTAx is positive definite.
  8. h. By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
  9. i. The largest value of a quadratic form xTAx, for ||x|| = 1, is the largest entry on the diagonal of A.
  10. j. The maximum value of a positive definite quadratic form xTAx is the greatest eigenvalue of A.
  11. k. A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable x = Pu, for some orthogonal matrix P.
  12. l. An indefinite quadratic form is one whose eigenvalues are not definite.
  13. m. If P is an n × n orthogonal matrix, then the change of variable x = Pu transforms xTAx  into a quadratic form whose matrix is P−1AP.
  14. n. If U is m × n with orthogonal columns, then UUT x is the orthogonal projection of x onto Col U.
  15. ○.      If B is m × n and x is a unit vector in ℝn , then ||Bx|| ≤ σ1, where σ1 is the first singular value of B.
  16. p. A singular value decomposition of an m × n matrix B can be written as B = PΣQ, where P is an m × n orthogonal matrix, Q is an n × n orthogonal matrix, and Σ is an m × n “diagonal” matrix.
  17. q. If A is n × n, then A and ATA have the same singular values.

a)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If A is orthogonally diagonalizable, then A is symmetric” is true or false.

Answer to Problem 1SE

a.

The given statement is true.

Explanation of Solution

Justification of statement istrue:

Theorem 2:

“An n×n matrix A is orthogonally diagonalizable if and only if A is a symmetric

matrix”

Refer Theorem 2.

The given statement “If A is orthogonally diagonalizable, then A is symmetric” is true.

Proof:

Consider A n×n matrix.

The matrix A is orthogonally diagonalizable if

A=PDPT=PDP1

Here, P is the orthogonal matrix such that P1=PT and D is the diagonal matrix.

Calculate the transpose of matrix A as follows:

AT=(PDPT)T=PTTDTPT=PDTPT=PDPT

Substitute A for PDPT .

AT=A

The matrix A and its transpose are equal.

The matrix A is symmetric.

b)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If A is an orthogonal matrix, then A is symmetric” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Consider an orthogonal matrix A as follows:

A=[0110]

Consider the transpose of matrix A is denoted by AT .

Show the transpose of the matrix A as follows:

AT=[0110]

Compare AT and A.

The matrix A and å AT are not equal. Then,

The matrix A is not symmetric.

Thus, the statement “If A is an orthogonal matrix, then A is symmetric” is false.

c)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If A is an orthogonal matrix, then Ax=x for all x in n ” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement is true:

Apply Theorem 7 of Section 6.2 as shown below.

“Let U be an m×n matrix with orthonormal columns, and let x and y be in Rn . Then

  1. a. Ux=x
  2. b. (Ux)(Uy)=xy
  3. c. (Ux)(Uy)=0 if and only if xy=0 .”

Calculate the value of (Ux)(Uy) as follows:

(Ux)(Uy)=(Ux)T(Uy)=xTUTUy

Here, UTU=I .

Substitute I for UTU .

(Ux)(Uy)=xTIy=xTy=xy (1)

Consider y=x .

Substitute x for y in Equation (1).

(Ux)(Ux)=xxUx2=x2Ux=x

Thus, The given statement “If A is an orthogonal matrix, then Ax=x for all x in n ” is true.

d.

Expert Solution
Check Mark
To determine

To mark:

The given statement “The principal axes of a quadratic form xTAx can be the columns of any matrix P that diagonalizes A” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Consider an orthogonal matrix P.

The principal axes of xTAx are the columns of the orthogonal matrix P that diagonalizes A.

Consider A has eigenvalue whose eigenspace has dimension greater than 1, the principal axes are not uniquely determined.

The given statement “The principal axes of a quadratic form xTAx can be the columns of any matrix P that diagonalizes A” is false.

e)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If P is an n×n matrix with orthogonal columns, then PT=P1 “ is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Apply definition of orthogonal set as shown below.

“A set of vectors {u1,...,up} in Rn is said to be an orthogonal set if each pair of distinct vectors from the set is orthogonal, that is, if uiuj=0 whenever ij .”

Apply definition of orthonormal set as shown below.

“A set {u1,...,up} is an orthonormal set if it is an orthogonal set of unit vectors.”

Consider a matrix P as follows:

P=[1111]

Consider the columns u=[11] and v=[11] .

Find the vectors are orthogonal as shown below.

uv=[11].[11]=1×(1)+1×1=1+1=0

Hence, the vector set is orthogonal.

Find the vectors are orthonormal as shown below.

u2=[11].[11]=1×1+1×1=1+1=2

Hence, the vector set is not orthonormal.

The given statement “If P is an n×n matrix with orthogonal columns, then PT=P1 “ is true or false.

f)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If every coefficient in a quadratic form is positive, thenthequadratic form is positive definite” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Refer Example 6 of Section 7.2.

The each terms in quadratic form Q is positive.

The quadratic form is not positive definite.

g)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If xTAx>0 for some x, then the quadratic form xTAx is positive definite” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Theorem 5.

“Let Abe an n×n symmetric matrix. Then a quadratic form xTAx is:

a. positive definite if and only if the eigenvalues of A are all positive,

b. negative definite if and only if the eigenvalues of A are all negative, or

c. indefinite if and only if A has both positive and negative eigenvalues.”

Consider a matrix A and x as follows:

A=[2003]x=[10]

The eigen value of the matrix A is 2 and 3 .

Calculate the value of xTAx as follows:

xTAx=[10][2003][10]=[10][2×1+00]=[10][20]=1×2+0

xTAx=2

Refer Part (c) of the Theorem 5.

The quadratic form xTAx is indefinite.

The value of xTAx>0

The given statement “If xTAx>0 for some x, then the quadratic form xTAx is positive definite” is false.

h)

Expert Solution
Check Mark
To determine

To mark:

The given statement “By a suitable change of variable, any quadratic form canbe hanged into one with no cross-product term” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement is true:

Theorem 4:

“Let Abe an n×n symmetric matrix. Then there is an orthogonal change of variable, x=Py , that transforms the quadratic form xTAx into a quadratic form yTDy with no cross-product term.”

Refer Theorem 4.

The given statement “By a suitable change of variable, any quadratic form can be hanged into one with no cross-product term” is true.

i)

Expert Solution
Check Mark
To determine

To mark:

The given statement “The largest value of a quadratic form xTAx , for x=1 ,is the largest entry on the diagonal of A” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Refer Example 3 of Section 7.3.

The given statement “The largest value of a quadratic form xTAx , for x=1 , is the largest entry on the diagonal of A” is false.

j)

Expert Solution
Check Mark
To determine

To mark:

The given statement “The maximum value of a positive definite quadratic form xTAx is the greatest eigenvalue of A” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

The maximum value must be computed over the set of unit vectors. Without a restriction on the norm of x , the value of xTAx can be made as large as possible.

Thus, the given statement “The maximum value of a positive definite quadratic form xTAx is the greatest eigenvalue of A” is false.

k)

Expert Solution
Check Mark
To determine

To mark:

The given statement “A positive definite quadratic form can be changed intoa negative definite form by a suitable change of variable x=Pu , for some orthogonal matrix P” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Consider the orthogonal change of variable x=Py . It changes a positive quadratic form into another quadratic form.

Refer Theorem 5.

The new quadratic form of xTAx is P1AP .

The value of P1AP is A.

Thus, it has the same eigen value same as A.

l)

Expert Solution
Check Mark
To determine

To mark:

The given statement “An indefinite quadratic form is one whose eigenvaluesare not definite” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

The term “definite eigenvalue” is undefined.

Thus, the term “definite eigenvalue” is meaningless.

m)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If P is an n×n orthogonal matrix, then the change of variable x=Pu transforms xTAx into a quadratic form whose matrix is P1AP ” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement is true:

Consider the value of x=Pu .

Calculate the value of xTAx as follows:

Substitute Pu for x.

xTAx=(Pu)TA(Pu)=uTPTAPu

Substitute P1 for PT .

xTAx=uTP1APu

n)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If U is m×n with orthogonal columns, then UUTx is the orthogonal projection of x onto Col U” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is false:

Consider the value of U=[1111] .

For UUTx be the orthogonal projection of x onto Col U, the columns of U must be orthonormal.

o)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If B is m×n and x is a unit vector in n , then Bxσ1 , where σ1 is the first singular value of B” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement is True:

Consider B be a n×n matrix.

Consider x is aunt matrix in n

Refer Example 2 and Example 1 of Section 7.4.

Get the value of Bxσ1

The σ1 is the first singular value of B.

p)

Expert Solution
Check Mark
To determine

To mark:

The given statement “A singular value decomposition of an m×n matrix B can be written as B=PΣQ , where P is an m×m orthogonal matrix, Q is an n×n orthogonal matrix, and is an m×n “diagonal” matrix” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Justification of statement is True:

Theorem 10:

“Let Abe an m×n matrix with rank r. Then there exists an m×n matrix as in (3) for which the diagonal entries in D are the first r singular values of A, σ1σ2σ3σr>0 , and there exist an m×m orthogonal matrix U and an n×n orthogonal matrix V such that

A=UΣVT

Refer Theorem10.

The matrices U and V are orthogonal. Then the matrix V is invertible and

V1=VT

Calculate the value of (VT)1 as follows:

(VT)1=(V1)1=V

The matrix V is invertible and square.

Then, VT is orthogonal matrix.

q)

Expert Solution
Check Mark
To determine

To mark:

The given statement “If A is n×n , then A and ATA have the same singular values” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Justification of statement is False:

Consider the matrix A=[2001] .

The singular values of matrix A are 2 and 1.

Consider the value of AT as follows:

AT=[2001]

Calculate the value of ATA as follows:

ATA=[2001][2001]=[4001]

The singular of ATA are 4 and 1.

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Chapter 7 Solutions

Linear Algebra and Its Applications (5th Edition)

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. 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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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