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Relationship between eigenvalues of 2x2 matrices within a 4x4 matrix

  1. Dec 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a 4 x 4 matrix A =
    B C
    0 D
    where B, C, and D are 2 x 2 matrices. What is the relationship between the eigenvalues of A, B, C, and D?

    3. The attempt at a solution
    I suppose you can write A as:
    b1 b2 c1 c2
    b3 b4 c3 c4
    0 0 d1 d2
    0 0 d3 d4

    and then find the eigenvalues of all of them in terms of those variables, but that seems like a lot of work that might not pay off.

    I know the characteristic equation for a 2x2 can be written (as follows) but I don't know what to do with it.

    λ^2 - trB λ + det B


    Thanks.
     
  2. jcsd
  3. Dec 6, 2011 #2

    lanedance

    User Avatar
    Homework Helper

    consider an eigenvector of B, u1 = (b1,b2)

    now lets extend that vector to be u1' = (b1,b2,0,0). How does the extended vector act on A?
     
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