https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/ Here is the proof I wrote, in Mathematica: ZS[n_] := M[n, n] (M[1, 1] + M[2, 2] + M[3, 3]) - M[n, n] (L1 + L2 + L3) ZS[n_, Li_] := Li (-M[n, n] - t @@ Complement[Range[3], {n}] + L1 + L2 + L3) ZS0 = Plus[M[1, 1] M[2, 2] - M[1, 2] M[2, 1], M[1, 1] M[3, 3] - M[1, 3] M[3, 1], M[2, 2] M[3, 3] - M[2, 3] M[3, 2], -(L1 L2 + L1 L3 + L2 L3)]; ZSComp[n_, Li_] := Subtract[ZS[n], ZS[n, Li] + ZS0] evsq[Li_] := With[{Ann = Complement[{L1, L2, L3}, {Li}]}, Part[Dot @@ ((Array[M, {3, 3}] - # IdentityMatrix[3]) & /@ Ann), #, #] & /@ Range[3]] Expand[evsq[L1] - (ZSComp[#, L1] & /@ Range[3])] // MatrixForm Expand[evsq[L2] - (ZSComp[#, L2] & /@ Range[3])] // MatrixForm Expand[evsq[L3] - (ZSComp[#, L3] & /@ Range[3])] // MatrixForm The so-called "new identity" is just the projector formula modulo a few zero-sums. The zero-sums are well known in the theory of matrix invariants, see for example: https://en.wikipedia.org/wiki/Invariants_of_tensors#Principal_invariants --Brad