/*! 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} Q29E Consider a diagonal n脳n聽matrix... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q29E (page 346)

Consider a diagonal n脳nmatrix Awith rank A=r<n. Find the algebraic and the geometric multiplicity of the eigenvalue 0of Ain terms of rand n

Short Answer

Expert verified

the algebraic and the geometric multiplicity of the eigenvalue is

almu(0) = n - r, gemu (0) = n - r

Step by step solution

01

Geometric multiplicity

Consider an eigenvalue 位of an n脳nmatrix A. The dimension of eigenspace E位=ker(A-位濒n)is called the geometric multiplicity of eigenvalue 位, denoted gemu(位). Thus,

gemu(位)=nullity(A-ln)=n-rank(A-ln)

gemu(l)=dim(E1)=1

02

Solution of the problem

Since rank A=r<n , we can assume that the first columns of form a basis for imA. Since the other n - r columns are linear combinations of the first r columns, this means that almu.

Lastly, we get gemu(0) = dimker(A - 0l)=dimker A = n-dimimA = n -r

Here, the result is

almu(0) = n - r, gemu(0) = n - r

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

[51-521082-7]

Is v鈬赌an eigenvector ofA+2In? If so, what is the eigenvalue?

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

For m鈮�n, find the dimension of the space of alln脳nmatricesfor which all the vectorse鈬赌1,....,e鈬赌mare eigenvectors.

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.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.