Eigenvalues of a linear transformation (Matrix)

  1. 1. The problem statement, all variables and given/known data
    Let T: M22 -> M22 be defined by

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


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


    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. jcsd
  3. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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?
  4. 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
  5. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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.
  6. 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.
  7. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

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