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

[brushman]

Homework Statement

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?

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 BThanks.

 

[lanedance]

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

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