# Eigenvectors and eigenspaces

1. Sep 8, 2011

### derryck1234

1. The problem statement, all variables and given/known data

Let T: M22 - M22 be defined by

T(a b = 2c a + c
c d) b-2c d

(a) Find the eigenvalues T.
(b) Find bases for the eigenspaces of T.

2. Relevant equations

det(lambdaI - A) = 0

3. The attempt at a solution

I just wanted to know how to start here. I am used to using vector transformations...

Would I use identity matrices 1 0 0 1 0 0 0 0
0 0 0 0 1 0 0 1

And then form a matrix somehow? I just don't know how big or how this matrix would be formed...?

2. Sep 8, 2011

### micromass

One small thing: you shouldn't call the matrices

$$\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right), \left(\begin{array}{cc} 0 & 1\\ 0 & 0\end{array}\right), \left(\begin{array}{cc} 0 & 0\\ 1 & 0\end{array}\right), \left(\begin{array}{cc} 0 & 0\\ 0 & 1\end{array}\right)$$

identity matrices. We reserve the term identity matrix for something else.
You can call them elementary matrices however.

Anyway, your approach is good: try to find the images of the elementary matrices to find the matrix form of T. T will have the form of a 4x4-matrix.

3. Sep 8, 2011

### derryck1234

Ok. So I find the matrix for T to be:

0 1 0 0
0 0 1 0
2 1 0 0
-2 0 0 1

The characteristic equation of which I find to be:

(lambda)4 - (lambda)3 - (lambda)2 - (lambda) + 2

Just wondered if there was an easy way to solve this?

4. Sep 8, 2011

### micromass

Hmm, that's not what I get:

$$T\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right)=\left(\begin{array}{cc} 0 & 1\\ 0 & 0\end{array}\right)$$

So the first column needs to be (0,1,0,0), no?

5. Sep 8, 2011

### derryck1234

O ok. So I must turn the resulting matrix into a column vector? I just put them together as matrices...my bad...ok I shall try this agen...shall get back to you to confirm my answer...thanks alot...

Ciao...I feel mentally ill that I am such a dumb ***!...but then again...have to learn somehow...

6. Sep 8, 2011

### derryck1234

Ok I think I have solved it:

The standard matrix for T is:

0 0 2 0
1 0 1 0
0 1 -2 0
0 0 0 1

Whose characteristic equation is:

(lambda - 1)2(lambda + 1)(lambda + 2) = 0

This right?

Thanks

Derryck

7. Sep 8, 2011

### micromass

Yes, that's also what I've got!!

8. Sep 9, 2011

### derryck1234

Thanks alot micromass

Tell you what if it wasn't for physicsforums I don't know how I would have passed my correspondence linear algebra course...

Cheers

Derryck