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

by td21
Tags: characteristic, polyno, simplest
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td21
#1
Jun11-11, 03:59 AM
td21's Avatar
P: 85
polynomial?
i find this but i do not understand;its too complex.

http://www.math.sc.edu/~howard/Classes/700/charAB.pdf

any simpler way/idea?Thanks!
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Simon_Tyler
#2
Jun11-11, 05:58 AM
P: 313
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)
feynman137
#3
Jun11-11, 07:16 AM
P: 8
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.


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