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Chapter 7: Q4E (page 345)

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

0-112

Short Answer

Expert verified

In the given matrix gemu (1) < almu (1). So the matrix is not diagonalizable.

Step by step solution

01

Algebraic versus.

Algebraic versus geometric multiplicity If 位 is an eigenvalue of a square matrix A,

then gemu(位) 鈮� almu(位).

det(A-位l)=0-位-112-位=0-位(2-位)+1=0

位2-2位+1=0位-12=0-位1,2=0

02

For, we solve

A-Ix=0-1-111x1x2=00-x1+-x2=0,x1+x2=0x1+x2=0

The basic of this eigenvector is1-1=v1

Therefore, gemu(1)< almu(1). So A is not diagonalizable.

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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.

[5-421]

If a vector v鈬赌is an eigenvector of both Aand B, isv鈬赌necessarily an eigenvector of A+B?

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Rotation through an angle of 180掳inR2.

find an eigenbasis for the given matrice and diagonalize:

A=[123246369]
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