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Chapter 7: Q60E (page 325)

find an eigenbasis for the given matrice and diagonalize:

A=19[74-4418-481]

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

Short Answer

Expert verified

The eigenbasis for the given matrice is 10001000-1.

Step by step solution

01

Solving the given matrices

Every vector v鈭�V, Where V is the given plane , will reflect onto itself, so we can choose two non-colinear eigenvector:

v1=210

And

v2=-210

With the eigenvalue位=1

Any vector perpendicular to V will project onto 0, so we can choose an eigrnvector

v3=1-22

With the eigenvalue位=-1

02

Solving further

Now,v1,v2,v3is an eigenbasis forR3, therefore the diagonalization of A in the eigenbasis is

10001000-1

Hence the final answer is10001000-1

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Most popular questions from this chapter

If v鈬赌 is an eigenvector of matrix A with associated eigenvalue 3 , show that v鈬赌is an image of matrix A .

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.

[3-25107002]

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.

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.

[-1-1-1-1-1-1-1-1-1]

25: Consider a positive transition matrix

A=[abcd]

meaning that a, b, c, and dare positive numbers such that a+ c= b+ d= 1. (The matrix in Exercise 24 has this form.) Verify that

[bc]and[1-1]

are eigenvectors of A. What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1?

Sketch a phase portrait.
See all solutions

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