# Norm of Eigenvectors from Eigenvalues

Nov 24th, 2019
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1. (*Norm of Eigenvectors from Eigenvalues*)
2. (*start*)
3. Clear[nn, t, n, k, M, x];
4. nn = 12;(*size of matrix*)
5. t[n_, 1] = 1;
6. t[1, k_] = 1;
7. t[n_, k_] :=
8. t[n, k] =
9. If[n < k,
10. If[And[n > 1, k > 1], -Sum[t[k - i, n], {i, 1, n - 1}], 0],
11. If[And[n > 1, k > 1], -Sum[t[n - i, k], {i, 1, k - 1}], 0]];
12.
13. "T is your favourite nn*nn Hermitian matrix"
14. T = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
15. HermitianMatrixQ[T]
16.
17. TableForm[A = Table[Table[T[[n, k]], {k, 1, nn}], {n, 1, nn}]];
18. lambdaA = N[Eigenvalues[A]];
19. TableForm[M = Table[Table[T[[n, k]], {k, 1, nn - 1}], {n, 1, nn - 1}]];
20. lambdaM = N[Eigenvalues[M]];
21. Quiet[Table[
22. N[Product[(lambdaA[[i]] - lambdaM[[k]]), {k, 1, nn - 1}]/
23. Product[If[Not[k == i], (lambdaA[[i]] - lambdaA[[k]]), 1], {k, 1,
24. nn}]], {i, 1, nn}]]
25. Table[Norm[N[Eigenvectors[A]][[i]]]^(-2), {i, 1, nn}]
26. "The values are equal and thereby the left hand side is equal to the \
27. right hand side in the formula."
28. (*end*)