Proof that the Trace is the Sum of the Eigenvalues

[iocal]

Hi all,

I recently came across that important result and I was wondering if someone could provide me with a proof. Please explain the steps though because I am a beginner.

Thank you!

 

[daon2]

Hi all,

I recently came across that important result and I was wondering if someone could provide me with a proof. Please explain the steps though because I am a beginner.

Thank you!

The proof is simple. Show that if A and B are similar matrices then they have the same trace. Put A into its jordan form J = P^{-1}AP. Then J has the same eigenvalues as A, and the eigenvalues of J are on the main diagonal. J and A are similar, so the trace of A is the sum of the eigenvalues of J is the sum of the eigenvalues of A