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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2Fb17c172a-f2b1-4cd5-899e-ecbe388bae51%2Fgrk4yod_processed.jpeg&w=3840&q=75)
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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