Are Negative Eigen Values of a Hessian Matrix physically acceptable? Suppose I have a Hessian Matrix of a System with 3N degrees of freedom, What are the physical significance of eigen values of the Hessian, Are negative Eigen Values physically acceptable?
 A: Assuming you are referring to the Hessian matrix $\mathbf{H}=\nabla\nabla U$ of the potential energy $U$ of a system (such as is used often in architecture and mechanics problems), then yes, it is possible for $\mathbf{H}$ to have negative eigenvalues.
But, it's usually a bad sign (no pun intended), and I'll explain why.
Recall what the eigenvalues of the Hessian are supposed to tell you about the system. Given some system in a $3N$-dimensional configuration space with potential energy $U(x_1,x_2,...,x_{3N})=U(\mathbf{x})$, the critical points $\mathbf{x}_\text{crit}$ of the configuration space satisfy $\nabla U(\mathbf{x}_\text{crit})=\mathbf{0}$, and in essence are the points where the system will, if placed there with zero velocity, remain stationary.
The question becomes, how do we know if $\mathbf{x}_\text{crit}$ is stable?
For example, you can place a pencil standing straight up on its tip, and if you do it carefully enough, it'll stay that way. But one tiny push will cause it to collapse. This is an example of unstable equilibrium. An example of stable equilibrium is a marble sitting at the bottom of a bowl; if you poke it, it'll roll around, but eventually it'll settle down again.
The sign of the eigenvalues of $\mathbf{H}=\nabla\nabla U(\mathbf{x}_\text{crit})$ tell you about stability. If an eigenvalue is positive, then a displacement of the system along the corresponding eigenmode will result in harmonic oscillations which will eventually damp down, leaving you in the same place as you started. 
But if you find a negative eigenvalue, then when you push the system in the direction of the corresponding eigenmode, instead of harmonic oscillation, it will exponentially accelerate in the direction of that eigenmode. This means that the system is unstable along that direction in configuration space, which is usually a bad thing, as it signifies that the structure will collapse if it moves in that direction.
In order for a system to be stable up to first order in the magnitude of the oscillations, all the eigenvalues of the Hessian should be nonnegative (ie, $\mathbf{H}$ should be positive semidefinite). If you have some negative eigenvalues, check your programming to see if there's an error present.
Also, be sure to check the magnitude of the negative eigenvalues; often zero eigenvalues will become perturbed by floating point errors during the computation, and usually have a magnitude on the order of machine precision. If your negative eigenvalues are that small, then it's probably nothing to worry about.
