Proof that the Trace of a Matrix is the sum of its Eigenvalues?

Question

I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.

Answer 1

I made a video for it here https://youtu.be/pXAf0Xyx0OM

Well, the problem with such a question is that you are dealing with a matrix that is nxn. So, you need to prove this for all matrices of size nxn. This type of proof generally requires a higher understanding of mathematics such as taking a proof course and a matrix theory course, which "generally" involves explaining it in plane English i.e. no arithmetic.

*I am curious as to what level of math you are at. Sometimes, you just need to prove this for a 3x3 and that is just a bit of tedious computation.

This question results in many cases i.e. is the matrix singular, diagonalizable, complex... so answering it in one quick line of arithmetic is not likely possible.

I think what would suit your pallet would be performing the proof for a diagonalizable matrix A

PROOF Let A be a diagonalizable matrix. Then, by 'Theorem of traces of matrix products,'

tr(A)=tr(SDS^(-1))=tr[(SD)S^(-1)]=tr[S^(-1)(SD)]=tr(ID)=tr(D). Q.E.D.

note: Recall that D is the diagonal matrix of eigenvalues.