Question

Show that (B^-1AB) has the same eigenvalues as (A). What is the relationship between the eigenvectors of (A) and (B^-1AB)?

a) They are orthogonal.
b) They are parallel.
c) They are identical.
d) They are perpendicular.

Answer 1

The matrix (B^-1AB) has the same eigenvalues as (A) and the eigenvectors are orthogonal.

Step-by-step explanation:

To show that (B-1AB) has the same eigenvalues as (A), we can use the property that similar matrices have the same eigenvalues. Since (B-1AB) is similar to (A), they will have the same eigenvalues.

The relationship between the eigenvectors of (A) and (B-1AB) depends on the transformation that matrix B-1 represents. If B-1 is an orthogonal matrix, then the eigenvectors will be orthogonal to each other. Therefore, the correct answer is a) They are orthogonal.