Apparent N-Dimensional Generalization (HACKWARNING): evsq[NDim_, Li_] := With[{Ann = Complement[L /@ Range[NDim], {Li}]}, Part[Dot @@ ((Array[M, {NDim, NDim}] - # IdentityMatrix[NDim]) & /@ Ann), #, #] & /@ Range[NDim]] v1[NDim_, ev_] := ((-1)^NDim)* Det[Array[M, {NDim, NDim}][[2 ;; -1, 2 ;; -1]] - L[1] IdentityMatrix[NDim - 1]] ZS[NDim_] := Reverse[Total[ (Det[Array[M, {NDim, NDim}][[#1, #1]]] &) /@ Subsets[Range[NDim], {#1}]] - Total[(Times @@ #1 &) /@ Map[L, Subsets[Range[NDim], {#1}], {2}]] & /@ Range[NDim - 1]] PolynomialMod[Expand[evsq[#, L[1]][[1]] + v1[#, L[1]]], ZS[#]] & /@ Range[2, 6] Out[]={0, 0, 0, 0, 0} etc... --Brad On Sat, Nov 30, 2019 at 1:21 PM Brad Klee <bradklee@gmail.com> wrote:
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