138k views
2 votes
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.

User AbdA
by
7.7k points

1 Answer

2 votes

Final answer:

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.

User Avitenberg
by
8.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.