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

In summary, the conversation discusses the relationship between the eigenvalues of a 4 x 4 matrix A, where B, C, and D are 2 x 2 matrices. The attempt at a solution suggests finding the eigenvalues of A and using the characteristic equation for a 2 x 2 matrix. Additionally, it proposes considering an eigenvector of B and extending it to act on A.
  • #1
brushman
113
1

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.
 
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  • #2
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?
 

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

1. What is the relationship between eigenvalues of 2x2 matrices and a 4x4 matrix?

The eigenvalues of a 4x4 matrix are the roots of the characteristic polynomial of the matrix. This polynomial is formed by taking the determinant of the matrix minus lambda times the identity matrix, and setting it equal to 0. The eigenvalues of the 2x2 matrices that make up the 4x4 matrix are the solutions to this polynomial.

2. How do I find the eigenvalues of a 2x2 matrix within a 4x4 matrix?

To find the eigenvalues of a 2x2 matrix within a 4x4 matrix, you can use the fact that the characteristic polynomial can be factored into the product of two 2x2 matrices. Set each of these matrices equal to 0 and solve for the eigenvalues, which will give you the eigenvalues of the 2x2 matrix within the 4x4 matrix.

3. Are the eigenvalues of a 2x2 matrix within a 4x4 matrix always real numbers?

No, the eigenvalues of a 2x2 matrix within a 4x4 matrix can be complex numbers. This is because the characteristic polynomial can have complex roots, which will result in complex eigenvalues.

4. What is the significance of the eigenvalues in a 4x4 matrix?

The eigenvalues of a 4x4 matrix represent the scaling factor of the corresponding eigenvectors. They also give information about the behavior of the matrix, such as whether it is invertible or not.

5. Can a 4x4 matrix have multiple sets of eigenvalues from different 2x2 matrices within it?

Yes, a 4x4 matrix can have multiple sets of eigenvalues from different 2x2 matrices within it. This is because a 4x4 matrix can be made up of several smaller matrices, each with their own eigenvalues. The overall eigenvalues of the 4x4 matrix will be the combination of these smaller matrices' eigenvalues.

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