Suppose A is diagonalizable with distinct eigenvalues...
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I will use the notation a1...ak for the eigenvalues (rather than using lambda, which is difficult to write).
I will also denote just by m the minimal polynomial of A. We know that because A is diagonalisable:
So P_j(t)=(t-a1)*...*(t-ak)/c_j where in the product the factor (t-aj) is missing, as it ...
This is a proof regarding a diagonalizable matrix and orthogonal projections.