What Are Eigenvalues and Eigenspaces in Linear Transformations?

[derryck1234]

Homework Statement

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.

Homework Equations

det(lambdaI - A) = 0

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...?

 

[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.

 

[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?

 

[micromass]

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?

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?

 

[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 a lot...

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

 

[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

 

[micromass]

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

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

 

[derryck1234]

Thanks a lot 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