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Chapter 7: Q61E (page 325)

find an eigenbasis for the given matrice and diagonalize:

A=17[6-2-3-23-6-3-6-2]

Representing the reflection about a plane E.

Short Answer

Expert verified

The eigenbasis for the given matrice is 10001000-1.

Step by step solution

01

Solving the given matrices

We can see that,

A100=67-27-37

So the plane is,

E=san13-2-3,30-1

02

Solving further

Every vector v鈭�Vwill reflect onto itself, so we can choose eigenvectors

v1=13-2-3

And

v2=30-1

With the eigenvalue位=1,

Any vector v perpendicular to V will reflect onto -v, so we can choose an eigenvector

v3=123

With the eigenvalue位=-1

03

Diagonalization

Now,v1.v2.v3is an eigenbasis for R3, therefore the diagonalization of A in the eigenbasis is

10001000-1

Hence the final answer is10001000-1

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