/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q7.6-31E Let A be a real 2 脳 2 matrix. S... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q7.6-31E (page 381)

Let A be a real 2 脳 2 matrix. Show that the zero state is a stable equilibrium of the dynamical systemx鈫�(t+1)=Ax鈫�(t)if (and only if)|trA|-1<detA<1

Short Answer

Expert verified

Proved|trA|-1<detA<1

Step by step solution

01

Define stability:

Stability is one of the most basic requirements for numerical models, which extends to mostly linear problems. If every solution that was initially close to it is always close to it. It is said to be stable and asymptotically stable although initially every solution close to it meets t鈫�鈭�.

02

Prove the given expression:

Consider if for both eigenvalues 位1,位2ofAwhich applies 位1<1,位2<1

So we get,detA=位1位2<1

Also we can apply,

位1,位2鈭�-1,1,-位1位2-1<位1+位2<位1位2+1鈬�-detA-1<trA<detA+1鈬�|trA|<detA+1Henceproved鈬�|trA|<detA+1

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

5.(45-2-2)

Find all2脳2matrices for whiche鈬赌1is an eigenvector.

Find a basis of the linear space V of all 2脳2matrices Afor which both[11]and[12]are eigenvectors, and thus determine the dimension of.

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[5-421]

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.