Let me re-iterate the previous objections (and a few more), which are not addressed in version 2. 1. Figure one has the unintended effect of showing the author's XX Century bias. 2. Obj. 1 is a serious problem considering the authors have missed an important proof. 2b. Given that symbolic computation is more rigorous, (see computer code below) the following case-by-case numerical computer code is not necessary: https://github.com/PeterDenton/Eigenvector-Eigenvalue-Identity 2c. I am aware of minor errors in defining null ring and zero sum from previous email. They are easy to fix and do not affect validity of Mma code. 3. Section 2.6 shows egocentric bias. If the strategy is proof by comparison, why not use the projector formula? It is easy and convincing, see Obj. 2&2b. 4. Section 3. The discovery of L34 is interesting, and suggestive that the adjective "unexpected" was only chosen out of ignorance. 5. The fact that L34 comes from a German author strongly supports the thesis that the identity should be viewed in terms of invariant theory ( a mostly German product ). 5b. Reference L34 should be investigated more closely to determine ties to invariant theory. 6. The authors do not discuss Cayley-Hamilton Thm. or its formalization by Frobenius, see also: http://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.ht... 7. The authors also did not mention JJ Sylvester; though, he was interested in the 3x3 case. 8. Section 5. If the authors want to discuss sociology, perhaps analyze bikeshed phenomena where three page note with four authors grows to 26 page ____ with lots of blog-comment coauthors (unlisted). 9. What about neutrinos? Is there such a thing as a neutrino galloping particle-wave? 10. Is "farbkreislaufphasenraumzeit" a proper word? 11. What should we do with all the extra horses in the lower-classes who don't get the big city P.R.? If Predrag C. thinks its worth working on, hopefully he will address some of these objections. I have copied him on this message. If you are looking for interesting math to read today, his book is also online:
See chapter 3. Save the birds ! ! ! We need the bird tracks ! ! ! --Brad On Wed, Dec 4, 2019 at 9:22 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Seems you were not alone ...
https://terrytao.wordpress.com/2019/12/03/eigenvectors-from-eigenvalues-a-su...
WFL
On 12/1/19, Brad Klee <bradklee@gmail.com> wrote:
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
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