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This is the first of the series of four videos on ways to diagonalize matrices. In this video, were gonna talk about traces and determinants. In successive videos, well talk about scaling and the identity, probability matrices, block matrices, a bunch of other techniques. So, for traces, the definition of the trace of a matrix is just the sum of the diagonal entries. A one, one plus A two, two plus A three, three and so on. And I claim, that if a matrix is diagonalizable, if you can write A as PDP inverse, then the trace of A is the same as the trace of D. And of course the matrix D has all the eigenvalues on the diagonal, and if you add them up, you get the sum of the eigenvalues. In other words, the trace of any matrix is the sum of the eigenvalues counted with multiplicity. If an eigenvalue has multiplicity three, you count it three times. Okay? Now, this formula turns out to work even if A is not diagonalizable. We have, havent really dealt with non-diagonal, how to deal with n