Representation by a diagonal matrix question

[zeion]

Homework Statement

Let T be the linear operator on R3 that has the given matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. a) Determine whether T can be represented by a diagonal matrix, and b) whenever possible, find a diagonal matrix and a basis of R3 such that T is represented by the diagonal matrix relative to the basis.

A = [tex]
\begin{bmatrix}8&5&-5\\5&8&-5\\15&15&-12 \end{bmatrix}



[/tex]

Homework Equations

The Attempt at a Solution

a) So first I find the eigenvalues I get {3, 3, -2}, then I check the eigenspace for each value and see that the dimension of the kernel corresponds to the multiplicity (3 has dimKer = 2, -2 has dimKer = 1), so then there is a matrix P such that A is diagonalized by P^-1AP. So the answer is Yes.

b) I don't really understand what it is asking me to do.. Can't I just use the eigenvectors as a basis for R3? I don't understand what this basis is I'm supposed to find and how to get T "represented by the diagonal matrix relative to the basis"..
What do I do with the given basis?

 

[VeeEight]

I haven't done this in a while, but I believe your first step is to find the change of basis matrix. This will then be your matrix P where you conjugate A as D=P^-1AP

 

[zeion]

What basis am I going from and to?

 

[VeeEight]

The basis you are going from is the one given on the first line of your question, {(1,0,0), (1,1,0), (1,1,1)}. The basis you are going to is the basis for R³ of eigenvectors of the matrix.

 

[vela]

b) I don't really understand what it is asking me to do.. Can't I just use the eigenvectors as a basis for R3?

Yes, that's exactly what you use. Part (a) didn't ask you to find a basis; it just asked you to determine if such a basis existed. Part (b) is asking you to find a basis that diagonalizes T.

 

[zeion]

Ok I'm kind of confused with these things:

I get a set of eigenvectors from A of T, I use them as a basis for T, and then I can use this set of eigenvectors written as the column of a matrix = P and when I do P^-1AP A will be diagonalized with the eigenvalues in the main diagonal?

So then if I get P from A wrt the basis A, then P is wrt to basis A?
So is it asking me to change P from wrt to A to A wrt to the eigenvectors basis?

 

[vela]

Ok I'm kind of confused with these things:

First, don't use the letter A to stand for two different things. It's confusing. Let's call A the matrix of T relative to basis U={(1,0,0), (1,1,0), (1,1,1)}.

I get a set of eigenvectors from A of T, I use them as a basis for T,

The eigenvectors aren't a basis for T; they're a basis for R³. The specific matrix representing T depends on what basis for R³ you're using.

and then I can use this set of eigenvectors written as the column of a matrix = P and when I do P^-1AP A will be diagonalized with the eigenvalues in the main diagonal?

Yes.

So then if I get P from A wrt the basis A, then P is wrt to basis A?

Think of P as a matrix which converts coordinates from the eigenvector basis to coordinates in the U basis. If you have some vector that has coordinates (x, y, z) in the eigenvector basis, you can get its coordinates in the U basis by multiplying P by (x, y, z). For example, if you multiply P by (1, 0, 0), which is the representation of the first eigenvector in the eigenvector basis, you get the first column of P, which is the eigenvector represented in the U basis.

So is it asking me to change P from wrt to A to A wrt to the eigenvectors basis?

No. The problem is simply asking you to find a basis in which the matrix for T is diagonal. As you know, that's the basis that consists of the eigenvectors of T. The only question left is how do you write down what those eigenvectors are. You could write them relative to the basis U (which is what you found when you found the eigenvectors using matrix A), the eigenvector basis, the canonical basis for R³, or any other basis for R³. Most likely, you're expected to express them relative to the canonical basis for R³.

 

[zeion]

So right now the eigenvectors are wrt to the U basis and it wants me to write them wrt to the canonical basis (standard basis?) of R3?