# What are the simplest way to show that AB and BA have the same characteristic polyno?

1. Jun 11, 2011

### td21

2. Jun 11, 2011

### Simon_Tyler

Re: what are the simplest way to show that AB and BA have the same characteristic pol

If you're happy that Det(AB)=Det(BA), then with I the identity matrix,
the characteristic polynomial is

p(x) = Det(AB-xI) = Det(IAB-xI2) = Det(B-1BAB-xB-1BI)
= Det(BABB-1-xBIB-1) = Det(BA-xI)

3. Jun 11, 2011

### feynman137

Re: what are the simplest way to show that AB and BA have the same characteristic pol

If A is invertible, than it is evident that AB and BA are similar matrices, therefore they have the same characteristic polynomial. Otherwise, we notice that the equation in lambda, det(A-lambda I)=0 has finitely many solutions. We can take epsilon such that, for all lambda 0<|lambda|<epsilon, A-lambda I is invertible. Therefore, (A-lambda I)B and B(A-lambda I) must have the same characteristic equation. Also, det(xI-(A-lambda I)B)=det(xI-B(A-lambda I)). For fixed x, each side of this equation is a polynomial in lambda, hence it is continuous. We can take lambda->0 and we are done.