# Linear Algebra (eigenvectors, eigenvalues, and orthogonal projections)

## Homework Statement

I am part way done with this problem.... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now.

W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it.

a) Find the basis of {v1, v2} for this subspace.

b) These basis vectors are basis vectors for what eigenvalue?

c) Show that n=<1,1,1> is orthogonal to {v1 ,v2}

d) Show that n is orthogonal to all of W

e) n is an eigenvector for T for what eigenvalue

f) Using matrix with eigenvectors and one for eigenvalues, find the standard matrix of T.

## The Attempt at a Solution

I am part way done with this problem.... I don't know how to solve part e or part f.

W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it.

a) Find the basis of {v1, v2} for this subspace.
I found the basis of W to be v1=<-1,1,0> and v2=<-1,0,1>

b) These basis vectors are basis vectors for what eigenvalue?
I found the eigenvalue to be zero after multiplying W with v1 and v2.

c) Show that n=<1,1,1> is orthogonal to {v1 ,v2}
I just took the dot product of n*v1 and n*v2 and both were zero

d) Show that n is orthogonal to all of W
My reasoning is that n is orthogonal to all of W because W is a linear combination of {v1 ,v2}

e) n is an eigenvector for T for what eigenvalue
??????

f) Using matrix with eigenvectors and one for eigenvalues, find the standard matrix of T.
??????

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lanedance
Homework Helper
so you need to find the matrix T, which projects any vector onto the plane W

lanedance
Homework Helper
actually for e) you don't need to find T first, what is the projection of n onto the plane? once you consider that the eigenvalue should be obvious

I am interested to see the answer to e) and f) myself... Landedance (or anybody) can you finish this example??

lanedance
Homework Helper
For b) the eigenvalue is not zero, as the vector is within the plane, the projection onto the plane will be the vector itself, so
T.v1=v1
From which u should be able to read off the eigenvalue

The reasoning behind a, c, d is ok

so why do you need a matrix with eigenvectors and one with eigenvalues to find the standard matrix T?

Is it a diagonalization problem where T=A and A=PDP^-1 where D is a matrix of the eigenvalues and P a matrix of the eigenvectors?

Is there a repeated eigenvalue? seems like T must be 3x3 since the basis, <-1,1,0> and <-1,0,1> are in R3 right? So there must be a 3rd vector? and a repeated eigenvalue?

Then you could use PDP^-1 to find A right?

Okay so if T is an othogonal projection onto W then the eigen values for {v1, v2} must be 1 then right?

And if n is orthogonal to W then T*(n) should be equal to zero right? That would make the eigenvalue for n equal to zero.

So to find the standard matrix of T I could set P=[n,v1,v2] which would then make the diagonal matrix D=[0,1,1]

I think that this is correct, but I am not sure.

(I am typing this on my iPod so I can't really put in the full matrices, but I am sure you get the idea.)

lanedance
Homework Helper
Okay so if T is an othogonal projection onto W then the eigen values for {v1, v2} must be 1 then right?
And if n is orthogonal to W then T*(n) should be equal to zero right? That would make the eigenvalue for n equal to zero.
So to find the standard matrix of T I could set P=[n,v1,v2] which would then make the diagonal matrix D=[0,1,1]

I think that this is correct, but I am not sure.

(I am typing this on my iPod so I can't really put in the full matrices, but I am sure you get the idea.)
Yes, if I understand ur matrix correctly, then u have the right idea and the eigenvalues are correct

Okay, I am at my computer so just to make sure we both understand each other...

$$P = \begin{pmatrix} 1 & -1 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}$$

$$D = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Since P-1TP=D
So then PDP-1=T

Then $$T = \begin{pmatrix} \frac{2}{3} & \frac{-1}{3} & \frac{-1}{3} \\ \frac{-1}{3} & \frac{2}{3} & \frac{-1}{3} \\ \frac{-1}{3} & \frac{-1}{3} & \frac{2}{3} \end{pmatrix}$$

lanedance
Homework Helper
it would be easier to normalise you eigenvectors first, as then P^{-1} = P^Tm but thats the idea

to check, see whether you your eigenvectors behave as you would expect

it would be easier to normalise you eigenvectors first, as then P^{-1} = P^Tm but thats the idea

to check, see whether you your eigenvectors behave as you would expect
What do you mean by P^Tm?

Is T in this sense a transpose or the matrix T? Also what is m?

Thank you for all your help so far it has really helped me clear up a lot of things!

lanedance
Homework Helper
sorry for the poor notation, but yes i meant the transpose, m was a mistype