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Chapter 7: Q62E (page 325)

find an eigenbasis for the given matrice and diagonalize:

A=114[13-2-3-210-6-3-65]

representing the orthogonal projection onto a plane E.

Short Answer

Expert verified

The eigenbasis for the given matrice is 100010000.

Step by step solution

01

Solving the given matrices:

We can see that,

A100=1314-17-314

So the plane is,

E=span13-2-3,30-1

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 0, so we can choose an eigrnvector

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 100010000.

Hence the final answer is 100010000.

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Most popular questions from this chapter

Find a 2脳2matrix A such that [31]and [12] are eigenvectors of A , with eigenvalues 5 and 10 , respectively.

Three holy men (let鈥檚 call them Anselm, Benjamin, and Caspar) put little stock in material things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to the two others, in equal parts. For example, if Anselm has 4 ounces one day, he will give 2 ounces each to Benjamin and Caspar.

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Reflection about a line L inR2.
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