@Robin,

your reference obviously assumes real matrices without notice  (this is a bad behavior found particularly in programming literature) and thus tells us only the well-known things.

The 2 by 2 compex symmetric matrix A with A11=A22=1, A12=A21=i has eigenvalues 1+i and 1-i which are not real!

Of course for 2x2 matrices and 3x3 matrices the conditions for real eigenvalues can easily read off from the explicit formulas for the eigen values. The results are complicated and don't look interesting.

@Biswanath

do you distribute teaser questions or do you need this information?

Actually  I notice that in case of 2x2 matrix  (difference between diagonal elements > 2tmes the non-diagonal element for real else complex . Using this what is the general condition in (mxm) matrix . Further I notice 

(1) H = p^2 + x^4 +3iX  spectrum is real but H=p^2 + x^4 + 7ix is complex  .

(ii) h = p^2 +ix^3 is real and complex but  H=p^2 + e^(ix^3) is real 

(3) H=P^2 + ix  is complex but H=p^2 + x^2 + ix is real .

I am yet to understand its relation with symmetric complex matrix eigenvalues .

I never mean any thing but I feel this examples point finger at symmetric matrix 

eigenvalue relation .

If you have the literature please give informations relating to that 

Thank you for earlier message .

Prof Rath

@Robin,

Thank you for confronting me with the Takagi factorization first in my live.

That for your E, M, U we have E = U M U^T is obviously true. What do you mean with 'acceptable' in such a context ? Do you intent to say something about my counter-example? Please comment on the intent of your remark.

@Ulrich

It is not a teaser. He has been working on Non-Hermitian Quantum theory. 

@Robin

The Takagi factorization is a surprising fact that obviously is relevant for Biswanath's question. I would like to leave the teaser with Biswanath to draw the consequences in the context of his investigations.

Prof Mutze

Do you find some answer to my problem. For your information I rarely use RG to solve

my problem. I had no  nor having now any intention to test others knowledge.

.  I always believe RG is a forum for helping each othet . However I hardly accept others

as long as I am not satisfied.

Prof Rath

@Biswanath,

sorry that you are not satisfied. As I pointed out already, I don't think that your original question has a canonical answer. For  growing size of the matrix the complexity of the condition grows too.  I'm convinced that Robin's pointer to the Takagi factorization is a gem for everybody working with symmetric complex matrices. I will not try to find out which of the obvious and less obvious properties of this gem may be useful for you. This is either your turn or the content of a new question.

A diagonal matrix with only real eigenvalues is Hermitian, then every matrix with real eigenvalues is Hermitian.  If it is symmetric too, it is real.

@Claude

A matrix with only real eigenvalues is of the form ADA^-1 with an invertible matrix A and a real diagonal matrix D. I don't see that such an ADA^-1 is always Hermitian. Rather, I'm quite sure that it is not difficult to construct a counter-example.

@Ulrich, so I'm sure you will provide one.  If A is unitary, or a multiple of a unitary matrix, ADA^-1 is Hermitian.

@Claude

Mathematica made the counter-example for me: A={{i,1},{i,i-1}} and D={{1,0},{0,2}} gives non-Hermitian ADA^-1. See the the appended file.

Symmetric tridiagonal complex  matrix can have only real eigenvalues. The matrix is not Hermitian.

@Ulrich, Ok.  A is actually not a multiple of a unitary matrix.

Dear Wiwat

Explain the spectra of H=p^2 + x^4 + i L x

L=2 ; spectra is real : L = 10 ; spectra is real and complex.

Do not talk about PT breaking. From math point of view using symmetric 

matrix , explain the spectra using your argument.

Any way I am not in a position to accept  your argument of trigonal matrix.

Eagerly waiting your reply .

Prof Rath

Dear Prof Rath,

Please choose a symmetric tridiagonal matrix which the spectra of H = p^2 + x^4 + i L x, L=2 ; spectra is real : L = 10.

Regards,

Wiwat Wanicharpichat

Dear Prof Rath,

I am sorry about my first answer, it is not true. For example: a complex symmetric tridiagonal matrix

[

1      1+i

1+i    2i

],

all eigenvalues of this matrix are 0, 1+2i   not real.

Regards,

Wiwat Wanicharpichat

Prof Damion

Thanks for giving two links.Regarding first link : my calculations partial agree with bender . In general I am not accepting the work of bender on ix^3. Regarding your

second link ;  it is of partial helpful as it support my findings. However the problem

remains still in doubt. Any way I am having a different a different lemma ; which can 

give the right answer, but people has to accept it . I am trying to put it in RG shortly.

Regards

Prof Rath

@Claude

 Although, as we have seen, not every matrix with real eigenvalues is Hermitian, it is true that every matrix with only real eigenvalues and a basis of orthogonal eigenvectors is Hermitian.

Yes, actually I assumed that the transformation matrix is orthogonal or unitary.  A diagonal matrix has orthogonal eigenvectors (with the usual inner product,) and this property is preserved by orthogonal/unitary transformation matrices.

@Ulrich

Can you suggest relation betweem diagonal and non-diagonal elements with real eigenvalues

Biswanath

@Biswanath,

unfortunately your question makes no sense to me, since it seems to speek of eigenvalues of matrix elements.

@Ulrich

When one calculates eigenvslue ,only matrix elements are considered. Take 2x2 ,3x3,4x4  matrix and rethink on my view point.. Further  give some time on 

spectra of H = p^2 +  x^4 + iLx 

withL= 2, 10 

One is pure real other is partially real . More ever Please give your specialisation:

Math ,Physics 

or Both . So that I will take your suggestions if necessary.

Prof Rath

@Biswanath

I'm not open to taking orders such as 'Further give some time on ...' and to decipher carelessly formulated messages.

Hey, looks like you don't know about matrix theory. Are you talking about N real eigenvalues, where N is dimension of your matrix? Because NxN matrix can have less then N eigenvalues and Jordan blocks. Can you purify your question?

@Eugene

(1) Matrix size NxN and symmetric

(2) Hence N eigenvalues alll real .

(3) Give the condition  

(4) Apply your method to justify the spectrum of 

(  i) H = p^2 + x^4 +2ix( all real  )

(  ii) H= p ^2 +x^4 +10ix(partially real and rest complex   )

I have taken N= 1400 ,2000 .

The symmetric matrix had all real diagonal elements. Non-diagonal elements some complex and some real .

Will you able to write the matrix ? If not use second quantisation.

If some problem write ,I send the matrix elements .So that you can

generate the matrix.

I eargerly wait for reply . If some clarification  required write .

Prof Rath 

@Eugene

You apply methods ,which you have known so far : answer the following

(i) H = p^2 + x^4 + 2 i x  (all real calculated about 1400 eigenvalues.(published work)

(ii) h = p^2  + x^4 + 10 i x  ( unpublished ) (partially real and complex)

waiting eagerly for your answer .Previously this question was not answered satisfactorily by many  people. If you say Jordan block ,then apply and give me the results . But do not  say the word PT breaking.

prof B .Rath