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I have a Matrix whose eigenvectors are needed to be calculated and then I have to calculate the Projector out of it.
But due to some unavoidable(atleast up to now), I am not able to carry out the procedure. 

(* Matrix definition *)
QEG = {{-(Exp[-I k]/Sqrt[2]), -((I Exp[-I k])/Sqrt[2])}, {-((I Exp[I k])/Sqrt[
   2]), -(Exp[I k]/Sqrt[2])}};

(* Matrix eigenvectors*)
EvQ = Eigenvectors[QEG];

(* Eigenvectors Normalization*)
NomQ = Simplify[Normalize[EvQ], Assumptions -> Element[k, Reals]];

(* Projectors*)
Projector1 = Simplify[KroneckerProduct[NomQ[[1]], Conjugate[NomQ[[1]]]], 
  Assumptions -> Element[k, Reals]] // MatrixForm
Projector2 = Simplify[KroneckerProduct[NomQ[[2]], Conjugate[NomQ[[2]]]], 
  Assumptions -> Element[k, Reals]] // MatrixForm

But the result for NomQ comes with Abs and the result of Projector1/2 comes with Conjugate. Couldn't seem to simplify it.

I went through enlightening answer(s) where they used ComplexExpand, it didn't work here, unfortunately, only made it worse.

L.K.
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  • NomQ = Assuming[Element[k, Reals], Normalize /@ EvQ // ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify] – Bob Hanlon Sep 28 '17 at 17:06
  • Thanks @BobHanlon, it worked but Projector1 still has those hanging Conjugate. Can I use this expression to replace them? It seems bit numerically expensive – L.K. Sep 29 '17 at 08:13
  • (Projector1 = Assuming[Element[k, Reals], KroneckerProduct[NomQ[[1]], Conjugate[NomQ[[1]]]]] // ComplexExpand[#, TargetFunctions -> {Re, Im}] & // Simplify) // MatrixForm Note that the MatrixForm formatting wrapper is isolated from the definition of Projector1 by the use of parentheses. – Bob Hanlon Sep 29 '17 at 13:50

1 Answers1

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The problem is that Normalize takes only vectors as input:

(*Matrix definition*)
QEG = {{-(Exp[-I k]/Sqrt[2]), -((I Exp[-I k])/
       Sqrt[2])}, {-((I Exp[I k])/Sqrt[2]), -(Exp[I k]/Sqrt[2])}};

(*Matrix eigenvectors*)
EvQ = Eigenvectors[QEG];

(*Eigenvectors Normalization*)
NomQ = 
 Table[Simplify[Normalize[QEG[[n]]], 
   Assumptions -> Element[k, Reals]], {n, 1, 2}];

(*Projectors*)
Projector1 = 
 Simplify[KroneckerProduct[NomQ[[1]], Conjugate[NomQ[[1]]]], 
   Assumptions -> Element[k, Reals]] // MatrixForm
Projector2 = 
 Simplify[KroneckerProduct[NomQ[[2]], Conjugate[NomQ[[2]]]], 
   Assumptions -> Element[k, Reals]] // MatrixForm

This gives:

$$ \left( \begin{array}{cc} \frac{1}{2} & -\frac{i}{2} \\ \frac{i}{2} & \frac{1}{2} \\ \end{array} \right)$$

and

$$\left( \begin{array}{cc} \frac{1}{2} & \frac{i}{2} \\ -\frac{i}{2} & \frac{1}{2} \\ \end{array} \right)$$

kiara
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  • Thanks Fabian. Just a little change I misquote QEG instead of EvQ, in normalize function(I edited my question), now comes the problem. So result may change. Sorry for that – L.K. Sep 28 '17 at 15:26