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Chapter 7: Q7-8E (page 383)

TRUE OR FALSE

8. If v鈫�is an eigenvector of A, then v鈫�must be an eigenvector of A3as well.

Short Answer

Expert verified

True, v is also an eigenvalue of A, with the eigenvalue 位3.

Step by step solution

01

Define eigenvalue

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by位.

Eigenvalues are the unique set of scalar values which are related to a set of linear equations that are most likely seen in matrix equations. On solving a characteristic equation, we get the roots whose another name for the eigenvectors. It is a non-zero vector that, after applying linear transformations, can only be altered by its scalar factor.

02

Explanation for the eigenvalue λ3

v is also an eigenvector of A, with the eigenvalue 位,

Av=位v

A3v=A2(位v)=位A2v=位A(位v)=位2Av=位3v

So, v is also an eigenvector of A, with the eigenvalue 位3.

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Most popular questions from this chapter

Consider the matrix A=|abbc| where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

For which2脳2 matrices A does there exist a nonzero matrix M Such that AM=MD,whereD=[2003] Give your answer in terms of eigenvalues of A.

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

8((110022003)

find an eigenbasis for the given matrice and diagonalize:

A=19[74-4418-481]

Representing the reflection about the planex-2y+2z=0.

Consider the matrix

A=[344-3]

a. Use the geometric interpretation of this transformation as a reflection combined with scaling to find the eigenvaluesA.

b. Find an eigen basis for A.

c. Diagonalize A .
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