Register to reply

Eigenvalues of a linear transformation (Matrix)

Share this thread:
Iconate
#1
Nov4-09, 10:11 PM
P: 21
1. The problem statement, all variables and given/known data
Let T: M22 -> M22 be defined by
T[itex]

\[ \left( \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right)\]

[/itex]=
[itex]

\[ \left( \begin{array}{cc}
2c & a+c \\
b-2c & d \\
\end{array} \right)\]

[/itex]

Find the eigenvectors of T



3. The attempt at a solution

My main question is, Which matrix am I using to compute my eigenvectors?
Do I need to compute a basis first?

Where this problem differs from my other questions is that I am no longer producing a matrix from my basis vectors which I use to create [T]B

Any insight would be great, thanks.
Phys.Org News Partner Science news on Phys.org
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
Dick
#2
Nov4-09, 10:20 PM
Sci Advisor
HW Helper
Thanks
P: 25,251
M22 is a four dimensional vector space, right? A basis is e1=[[1,0],[0,0]], e2=[[0,1],[0,0]], e3=[[0,0],[1,0]] and e4=[[0,0],[0,1]], right? So T must be a 4x4 matrix in that basis, yes? Can you write out what it is in the {e1,e2,e3,e4} basis?
Iconate
#3
Nov4-09, 10:53 PM
P: 21
Yeah I did that, do I find the eigenvectors with each of their matricies?, normally id put my basis vectors INTO a matrix, but I have matricies, i figure they have to go somewhre, just dont know where

Dick
#4
Nov4-09, 11:00 PM
Sci Advisor
HW Helper
Thanks
P: 25,251
Eigenvalues of a linear transformation (Matrix)

I'm not sure I understand that. Just pretend [[a,b],[c,d]] is a 4 vector, [a,b,c,d]. T maps it to another 4 vector [2c,a+c,b-2c,d]. The fact they write these vectors as matrices is just a technicality.
Iconate
#5
Nov5-09, 02:29 PM
P: 21
I figured it out. I have to write T[e1] as a linear combination of the basis vectors.

Ex. T(e1) = [[0,1],[0,0]] = 0*e1 + 1*e2 + 0*e3 + 0*e4
= (0,1,0,0)

And Now i have my vector! Computing this for all ei's will create my matrix P.
Dick
#6
Nov5-09, 02:39 PM
Sci Advisor
HW Helper
Thanks
P: 25,251
Quote Quote by Iconate View Post
I figured it out. I have to write T[e1] as a linear combination of the basis vectors.

Ex. T(e1) = [[0,1],[0,0]] = 0*e1 + 1*e2 + 0*e3 + 0*e4
= (0,1,0,0)

And Now i have my vector! Computing this for all ei's will create my matrix P.
Exactly.


Register to reply

Related Discussions
Linear transformation and its matrix Linear & Abstract Algebra 3
Eigenvalues/eigenvectors and linear transformation Calculus & Beyond Homework 2
The Matrix Of A Linear Transformation Calculus & Beyond Homework 3
Matrix of linear transformation Introductory Physics Homework 5
Matrix rep. of Linear Transformation Linear & Abstract Algebra 1