Proving that AB and BA have the same eigenvalues is important because it allows us to simplify the process of finding eigenvalues for a given matrix. This also allows us to make connections between the properties of AB and BA and use them to solve more complex problems.
The most common method for proving that AB and BA have the same eigenvalues is by using the fact that for any two square matrices, the determinant of their product is equal to the product of their determinants. Using this property, we can show that the characteristic polynomial of AB and BA are equal, which means they have the same eigenvalues.
Yes, we can take the matrices A = [1 2; 3 4] and B = [5 6; 7 8]. Both of these matrices have the eigenvalues 5 and 10, which can be easily verified by finding their characteristic polynomials.
Yes, there are some exceptions. For non-square matrices, it is possible for AB and BA to have different eigenvalues. Additionally, if the matrices A and B do not commute, meaning AB ≠ BA, then they may not have the same eigenvalues.
Proving that AB and BA have the same eigenvalues can be useful in solving a variety of real-world problems, such as in physics, engineering, and economics. It can help us understand the behavior of complex systems and make predictions based on the properties of the system's matrices.