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Chapter 7: Q5E (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

5.(45-2-2)

Short Answer

Expert verified

In the given matrix there is no real eigenvalues.

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).

detA-位l=04-位5-2-2=04-位-2-位+10=0位2-2位+2=0

02

Taking square root.

位2-2位+2=0位1,2=2卤4-82位1,2=1卤i

Therefore, the given matrix has no real eigenvalues.

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Most popular questions from this chapter

Find a basis of the linear space Vof all 2脳2matrices Afor whichrole="math" localid="1659530325801" [01]is an eigenvector, and thus determine the dimension of V.

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

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Is v鈬赌an eigenvector ofA-1? If so, what is the eigenvalue?

Find all 4x4matrices for whiche鈬赌2is an Eigen-vector.
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