Question

Consider two n×n matrices S and T. Show that, if at least one of the matrices are invertible, then ST and TS have the same set of real eigenvalues.

Answer 1

If at least one of the matrices, S and T, is invertible, then ST and TS will have the same set of real eigenvalues.

Step-by-step explanation:

If at least one of the matrices, S and T, is invertible, then ST and TS will have the same set of real eigenvalues.

To prove this, let v be an eigenvector of ST with eigenvalue λ. This means that STv = λv. Since S is invertible, we can rewrite this equation as T(Sv) = λ(Sv).

Let u = Sv. Then, we have Tu = λu. This shows that u is an eigenvector of TS with eigenvalue λ. Therefore, ST and TS have the same set of real eigenvalues.