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Chapter 7: Q16E (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

16(1100-1-1220)

Short Answer

Expert verified

The given matrix of A is not diagonalizable.

Step by step solution

01

Algebraic Versus.

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

then gemu(1)<almu(1)

detA-位I=01-位100-1-位-122-位=0-位(1-位)(-1-位)-2+2(1+位)=0-位(位2+1)=0位1=0,位2=-i,位3=i

02

Find the eigenvalues.

Here, the only real eigenvalues is 位=0

Ax=01100-1-1220x1x2x3=000x1+x2=0,x2+x3=0

The basic for eigenspace is1-11=v1

Therefore, there is only one real eigenvalues, A is not a diagonalizable in 鈩�3.

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Most popular questions from this chapter

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.

Orthogonal projection onto a line L inR3.

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

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?

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

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