Explanation Given information: Let A be an n × n symmetric matrix , let M and m denote the maximum and minimum values of the quadratic form x T A x ; where x T x = 1 , and denote corresponding unit eigenvectors by u 1 and u n . The following calculations show that given any number t between M and m , there is a unit vector x such that t = x T A x . Explanation: Refer to Equation (2) in the Section 7.3; m = min { x T A x : ‖ x ‖ = 1 } , M = max { x T A x : ‖ x ‖ = 1 } (1) Show the Theorem 6: Let A be a symmetric matrix, and define m and M as in Equation (1). Then M is the greatest eigenvalue λ 1 of A and m is the least eigenvalue of A . The value of x T A x is M when x is a unit eigenvector u 1 corresponding to M . The value of x T A x is m when x is a unit eigenvector corresponding to m . Consider the values of m = M . So, α = 0 . Determine the value of t ; t = ( 1 − 0 ) m + 0 M = m Determine the value of x ; x = 1 − ( 0 ) u n + ( 0 ) u 1 = u n Refer to Theorem 6; u n T A u n = m . Consider the values m < M and the value of t is in between m and M . 0 ≤ t − m ≤ M − m 0 ≤ ( t − m ) / ( M − m ) ≤ 1 Consider α = t − m M − m . The vectors 1 − α u n and α u 1 are orthogonal vectors since they are giving different eigenvectors for different eigenvalues

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ISBN: 9781292092232

Author: Lay, David C. (author.)

Publisher: PEARSON

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Chapter 7.3, Problem 13E

To determine

To verify: that t=(1−α)m+αM for some number α between 0 and 1. Then let x=1−αun+αu1 , and show that xTx=1 and xTAx=t .

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Chapter 7.3, Problem 12E

Chapter 7.3, Problem 14E

Chapter 7 Solutions

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 - a. 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To verify: that t = ( 1 − α ) m + α M for some number α between 0 and 1. Then let x = 1 − α u n + α u 1 , and show that x T x = 1 and x T A x = t .

Solution Summary: The author shows the Theorem 6: Let A be a symmetric matrix, and define m and M as in Equation (1).

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To verify: that t=(1−α)m+αM for some number α between 0 and 1. Then let x=1−αun+αu1 , and show that xTx=1 and xTAx=t .

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Chapter 7 Solutions