First, let's be clear about what you can do and what you can't do.  The characteristic equation for an 8x8 matrix to determine the 8 eigenvalues is in general an eighth order polynomial.  First you may want to read about Abel's Impossibility Theorem ( see: http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html ) for finding formulas for computing eigenvalues from polynomials. In particular see the link for quintic polynomials (http://mathworld.wolfram.com/QuinticEquation.html)  . So in general the only hope you have for a 8th order polynomial is if the substitution z=y^2 reduces the 8th order polynomial in y to a 4th order polynomial in z, which would then allow you to find an analytic expression for the eigenvalues.  In your case this is not possible as the coefficients (B, g1, g2, k1, k2, J) are not specified.

Nevertheless,  using a software packages ( here  I will discuss using Mathematica) you can readily explore the solution space for the eigenvalues. My preliminary calculations shows that the eigenvalues for random real numbers that I have selected  for parameter values between  (-2, 2) and  (0,5) give  eight real eigenvalues: 4 positive and 4 negative.

I have attached a PDF of a Mathematica notebook that illustrates these calculations. Enjoy 

dear dr chand

please check the attachment if it is helpful for you.

Dear Mohon Sir,

Main problem is to find generalised form. As elements are not constant, they are all variable. 

The eight eigenvalues of your matrix are the same and can be found by running the following code in Matlab:

syms B g2 k2 g1 k1 J g2

A=[2*B 0 g2 k2 g1 k1 2*J 0;

0 2*B k2 g2 0 g1 0 2*J;

g2 k2 0 0 2*J 0 g1 k1;

k2 g2 0 0 0 2*J 0 g1;

g1 0 2*J 0 0 0 g2 k2;

k1 g1 0 2*J 0 0 0 g1;

2*J 0 g1 0 g2 0 -2*B 0;

0 2*J k1 g1 k2 g1 0 -2*B];

eig(A)

The output, however, is not given in an analytic form.

Regards,

George

Dear George Sir,

Thanks. I am doing the same. But I need 8 particular expressions for these eigenvalues, which I have to used in farther calculation.It shows a 8 power polynomial expression. From that, I am not able to simply further. Can it be possible to simply further from that polynomial form and get some exact expression?

 First, let's be clear about what you can do and what you can't do.  The characteristic equation for an 8x8 matrix to determine the 8 eigenvalues is in general an eighth order polynomial.  First you may want to read about Abel's Impossibility Theorem ( see: http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html ) for finding formulas for computing eigenvalues from polynomials. In particular see the link for quintic polynomials (http://mathworld.wolfram.com/QuinticEquation.html)  . So in general the only hope you have for a 8th order polynomial is if the substitution z=y^2 reduces the 8th order polynomial in y to a 4th order polynomial in z, which would then allow you to find an analytic expression for the eigenvalues.  In your case this is not possible as the coefficients (B, g1, g2, k1, k2, J) are not specified.

Nevertheless,  using a software packages ( here  I will discuss using Mathematica) you can readily explore the solution space for the eigenvalues. My preliminary calculations shows that the eigenvalues for random real numbers that I have selected  for parameter values between  (-2, 2) and  (0,5) give  eight real eigenvalues: 4 positive and 4 negative.

I have attached a PDF of a Mathematica notebook that illustrates these calculations. Enjoy 

Thank you Very much Brain Sir. I have not expect such well explained answer. Thank you  for giving your valuable time and effort for this problem. 

At present I try to solve this, by reducing the parameters like g1=g2=0 and k1=k2. So effectively, I have now 3 parameters k, J, B. Also I try to rearrange the matrix in block form. Let's see. If I have face further problem, I will contact you.