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9. Let A € Rn be symmetric and A₁, A2, ...,An be its eigenvalues arranged such that Show that A₁2₂2.... *** A₁ = = max xAx. ||x||₂=1

9. Let A € Rn be symmetric and A₁, A2, ...,An be its eigenvalues arranged such that Show that A₁2₂2.... *** A₁ = = max xAx. ||x||₂=1

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 10AEXP
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**Eigenvalue Properties of Symmetric Matrices** **Problem Statement**: Let \( A \in \mathbb{R}^{n \times n} \) be a symmetric matrix and let \( \lambda_1, \lambda_2, \ldots, \lambda_n \) be its eigenvalues arranged such that \( \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n \). Show that: \[ \lambda_1 = \max_{\|x\|_2 = 1} x^T A x. \] **Explanation**: Given a symmetric matrix \( A \), its eigenvalues can be organized in a non-increasing order. The problem requires proving that the largest eigenvalue, \( \lambda_1 \), can be characterized as the maximum value of the quadratic form \( x^T A x \), where \( x \) is a vector constrained to have a Euclidean norm ( \( \|x\|_2 \) ) of 1. This principle is a fundamental result in linear algebra, particularly in the study of symmetric matrices, and leads to various applications in numerical methods and optimization.
Transcribed Image Text:**Eigenvalue Properties of Symmetric Matrices** **Problem Statement**: Let \( A \in \mathbb{R}^{n \times n} \) be a symmetric matrix and let \( \lambda_1, \lambda_2, \ldots, \lambda_n \) be its eigenvalues arranged such that \( \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n \). Show that: \[ \lambda_1 = \max_{\|x\|_2 = 1} x^T A x. \] **Explanation**: Given a symmetric matrix \( A \), its eigenvalues can be organized in a non-increasing order. The problem requires proving that the largest eigenvalue, \( \lambda_1 \), can be characterized as the maximum value of the quadratic form \( x^T A x \), where \( x \) is a vector constrained to have a Euclidean norm ( \( \|x\|_2 \) ) of 1. This principle is a fundamental result in linear algebra, particularly in the study of symmetric matrices, and leads to various applications in numerical methods and optimization.
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