Explanation Proof: Now we will show that for any orthogonal matrix B , ‖ B u ‖ = ‖ v ‖ ‖ B v ‖ 2 = ( B v , B v ) = ( B v ) t ( B v ) = v t B t B v =v t I n v [ Since B is an orthogonal matrix ] = v t v = ( v , v ) = ‖ v ‖ 2 Thus, we have ‖ B v ‖ = ‖ v ‖ for any vector v. Consider the singular value decomposition of A as A = U Σ V t , where U , V are orthogonal matrix. Then we have ‖ A v ‖ = ‖ ( U Σ V t ) v ‖ = ‖ U ( Σ V t v ) ‖ = ‖ Σ V t v ‖ [ Since U is an orthogonal matrix ] Now let, σ 1 , σ 2 , ... , σ m are singular values of A , where σ 1 is the largest and σ m is the smallest singular value. Then we have Σ = [ σ 1 0 … 0 0 σ 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ σ m 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 0 ] Since σ 1 is the largest and σ 2 is the smallest singular value, hence we have ‖ [ σ m 0 … 0 0 σ m ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0

Linear Algebra with Applications (2-Download)

5th Edition

ISBN: 9780321796974

Author: Otto Bretscher

Publisher: PEARSON

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Chapter 8.3, Problem 26E

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To show: the given expression σm‖v→‖≤‖Av→‖≤σ1‖v→‖ and compare with σ2≤‖Au→‖≤σ1 .

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Chapter 8.3, Problem 25E

Chapter 8.3, Problem 27E

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Chapter 8 Solutions

Linear Algebra with Applications (2-Download)

Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - Let L from R3 to R3 be the reflection about the...Ch. 8.1 - Consider a symmetric 33 matrix A with A2=I3 . Is...Ch. 8.1 - In Example 3 of this section, we diagonalized the...Ch. 8.1 - If A is invertible and orthogonally...Ch. 8.1 - Find the eigenvalues of the matrix...Ch. 8.1 - Use the approach of Exercise 16 to find the...Ch. 8.1 - Consider unit vector v1,...,vn in Rn such that the...Ch. 8.1 - Consider a linear transformation L from Rm to Rn ....Ch. 8.1 - Consider a linear transformation T from Rm to Rn ,...Ch. 8.1 - Consider a symmetric 33 matrix A with eigenvalues...Ch. 8.1 - Consider the matrix A=[0200k0200k0200k0] , where k...Ch. 8.1 - If an nn matrix A is both symmetric and...Ch. 8.1 - Consider the matrix A=[0001001001001000] . Find an...Ch. 8.1 - Consider the matrix [0000100010001000100010000] ....Ch. 8.1 - Let Jn be the nn matrix with all ones on the...Ch. 8.1 - Diagonalize the nn matrix (All ones along both...Ch. 8.1 - Diagonalize the 1313 matrix (All ones in the last...Ch. 8.1 - Consider a symmetric matrix A. If the vector v is...Ch. 8.1 - Consider an orthogonal matrix R whose first column...Ch. 8.1 - True or false? If A is a symmetric matrix, then...Ch. 8.1 - Consider the nn matrix with all ones on the main...Ch. 8.1 - For which angles(s) can you find three distinct...Ch. 8.1 - For which angles(s) can you find four distinct...Ch. 8.1 - Consider n+1 distinct unit vectors in Rn such that...Ch. 8.1 - Consider a symmetric nn matrix A with A2=A . Is...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - Show that for every symmetric nn matrix A, there...Ch. 8.1 - Find a symmetric 22 matrix B such that...Ch. 8.1 - For A=[ 2 11 11 11 2 11 11 11 2 ] find a nonzero...Ch. 8.1 - Consider an invertible symmetric nn matrix A. When...Ch. 8.1 - We say that an nnmatrix A is triangulizable if A...Ch. 8.1 - a. Consider a complex upper triangular nnmatrix U...Ch. 8.1 - Let us first introduce two notations. For a...Ch. 8.1 - Let U0 be a real upper triangular nn matrix with...Ch. 8.1 - Let R be a complex upper triangular nnmatrix with...Ch. 8.1 - Let A be a complex nnmatrix that ||1 for all...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - If A is a symmetric matrix, what can you say about...Ch. 8.2 - Recall that a real square matrix A is called skew...Ch. 8.2 - Consider a quadratic form q(x)=xAx on n and a...Ch. 8.2 - If A is an invertible symmetric matrix, what is...Ch. 8.2 - Show that a quadratic form q(x)=xAx of two...Ch. 8.2 - Show that the diagonal elements of a positive...Ch. 8.2 - Consider a 22 matrix A=[abbc] , where a and det A...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - a. Sketch the following three surfaces:...Ch. 8.2 - On the surface x12+x22x32+10x1x3=1 find the two...Ch. 8.2 - Prob. 23E Ch. 8.2 - Consider a quadratic form q(x)=xAx Where A is a...Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Consider a quadratic form q(x)=xAx , where A is a...Ch. 8.2 - Show that any positive definite nnmatrix A can be...Ch. 8.2 - For the matrix A=[8225] , write A=BBT as discussed...Ch. 8.2 - Show that any positive definite matrix A can be...Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Prob. 44E Ch. 8.2 - Prob. 45E Ch. 8.2 - Prob. 46E Ch. 8.2 - Prob. 47E Ch. 8.2 - Prob. 48E Ch. 8.2 - Prob. 49E Ch. 8.2 - Prob. 50E Ch. 8.2 - What are the signs of the determinants of the...Ch. 8.2 - Consider a quadratic form q. If A is a symmetric...Ch. 8.2 - Consider a quadratic form q(x1,...,xn) with...Ch. 8.2 - If A is a positive semidefinite matrix with a11=0...Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.2 - Prob. 57E Ch. 8.2 - Prob. 58E Ch. 8.2 - Prob. 59E Ch. 8.2 - Prob. 60E Ch. 8.2 - Prob. 61E Ch. 8.2 - Prob. 62E Ch. 8.2 - Prob. 63E Ch. 8.2 - Prob. 64E Ch. 8.2 - Prob. 65E Ch. 8.2 - Prob. 66E Ch. 8.2 - Prob. 67E Ch. 8.2 - Prob. 68E Ch. 8.2 - Prob. 69E Ch. 8.2 - Prob. 70E Ch. 8.2 - Prob. 71E Ch. 8.3 - Find the singular values of A=[1002] .Ch. 8.3 - Let A be an orthogonal 22 matrix. Use the image of...Ch. 8.3 - Let A be an orthogonal nn matrix. Find the...Ch. 8.3 - Find the singular values of A=[1101] .Ch. 8.3 - Find the singular values of A=[pqqp] . Explain...Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - If A is an invertible 22 matrix, what is the...Ch. 8.3 - If A is an invertible nn matrix, what is the...Ch. 8.3 - Consider an nm matrix A with rank(A)=m , and a...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Consider the standard matrix A representing the...Ch. 8.3 - Consider an SVD A=UVT of an nm matrix A. Show that...Ch. 8.3 - If A is a symmetric nn matrix, what is the...Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Show that any matrix of rank r can be written as...Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - For which square matrices A is there a singular...Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8 - The singular values of any diagonal matrix D are...Ch. 8 - Prob. 2E Ch. 8 - Prob. 3E Ch. 8 - Prob. 4E Ch. 8 - Prob. 5E Ch. 8 - Prob. 6E Ch. 8 - The function q(x1,x2)=3x12+4x1x2+5x2 is a...Ch. 8 - Prob. 8E Ch. 8 - If matrix A is positive definite, then all the...Ch. 8 - Prob. 10E Ch. 8 - Prob. 11E Ch. 8 - Prob. 12E Ch. 8 - Prob. 13E Ch. 8 - Prob. 14E Ch. 8 - Prob. 15E Ch. 8 - Prob. 16E Ch. 8 - Prob. 17E Ch. 8 - Prob. 18E Ch. 8 - Prob. 19E Ch. 8 - Prob. 20E Ch. 8 - Prob. 21E Ch. 8 - Prob. 22E Ch. 8 - If A and S are invertible nn matrices, then...Ch. 8 - Prob. 24E Ch. 8 - Prob. 25E Ch. 8 - Prob. 26E Ch. 8 - Prob. 27E Ch. 8 - Prob. 28E Ch. 8 - Prob. 29E Ch. 8 - Prob. 30E Ch. 8 - Prob. 31E Ch. 8 - Prob. 32E Ch. 8 - Prob. 33E Ch. 8 - Prob. 34E Ch. 8 - Prob. 35E Ch. 8 - Prob. 36E Ch. 8 - Prob. 37E Ch. 8 - Prob. 38E Ch. 8 - Prob. 39E Ch. 8 - Prob. 40E Ch. 8 - Prob. 41E Ch. 8 - Prob. 42E Ch. 8 - Prob. 43E Ch. 8 - Prob. 44E Ch. 8 - Prob. 45E Ch. 8 - Prob. 46E Ch. 8 - Prob. 47E Ch. 8 - Prob. 48E Ch. 8 - Prob. 49E Ch. 8 - Prob. 50E Ch. 8 - Prob. 51E Ch. 8 - Prob. 52E Ch. 8 - Prob. 53E Ch. 8 - Prob. 54E

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the given expression σ m ‖ v → ‖ ≤ ‖ A v → ‖ ≤ σ 1 ‖ v → ‖ and compare with σ 2 ≤ ‖ A u → ‖ ≤ σ 1 .

Solution Summary: The author analyzes the singular value decomposition of A as A=USigma Vt, where U, V are orthogonal matrix.

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To show: the given expression σm‖v→‖≤‖Av→‖≤σ1‖v→‖ and compare with σ2≤‖Au→‖≤σ1 .

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Chapter 8 Solutions