31. (a) Let A be an n × n-symmetric matrix. Diagonalize A to show that x • Ax || x || ² is greater than or equal to the smallest eigenvalue of A for all x # 0 in R". (b) Show that the quadratic form Q₁(x) = = x. Ax is coercive if and only if A is positive definite. (c) Conclude from (b) that if f(x) = a + b x + x Ax . • is any quadratic function where a € R, be R" and A is an n - n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.

31. (a) Let A be an n × n-symmetric matrix. Diagonalize A to show that x • Ax || x || ² is greater than or equal to the smallest eigenvalue of A for all x # 0 in R". (b) Show that the quadratic form Q₁(x) = = x. Ax is coercive if and only if A is positive definite. (c) Conclude from (b) that if f(x) = a + b x + x Ax . • is any quadratic function where a € R, be R" and A is an n - n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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31. (a) Let A be an n × n-symmetric matrix. Diagonalize A to show that X. Ax || x || ² is greater than or equal to the smallest eigenvalue of A for all x # 0 in R". (b) Show that the quadratic form Q₁(x) = x Ax is coercive if and only if A is positive definite. • (c) Conclude from (b) that if f(x) = a + b⋅x + 1x• Ax is any quadratic function where a ≤ R, b ≤ Rª and A is an n × n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.
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