Eigenvalues of a linear transformation (Matrix)by Iconate Tags: eigenvalues, linear, matrix, transformation 

#1
Nov409, 10:11 PM

P: 21

1. The problem statement, all variables and given/known data
Let T: M_{22} > M_{22} be defined by T[latex] \[ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\] [/latex]= [latex] \[ \left( \begin{array}{cc} 2c & a+c \\ b2c & d \\ \end{array} \right)\] [/latex] 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. 



#2
Nov409, 10:20 PM

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P: 25,174

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?




#3
Nov409, 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




#4
Nov409, 11:00 PM

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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,b2c,d]. The fact they write these vectors as matrices is just a technicality.




#5
Nov509, 02:29 PM

P: 21

I figured it out. I have to write T[e_{1}] as a linear combination of the basis vectors.
Ex. T(e_{1}) = [[0,1],[0,0]] = 0*e_{1} + 1*e_{2} + 0*e_{3} + 0*e_{4} = (0,1,0,0) And Now i have my vector! Computing this for all e_{i}'s will create my matrix P. 



#6
Nov509, 02:39 PM

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