If A is an nxn matrix, then it will have n eigenvalues, may be real or complex and may be distinct or not distinct. The modes are eigenvectors corresponding to eigenvalues. The eigenvectors are linearly independent. So the space spanned by these eigenvectors will be a subspace of Rn or Cn. The problem will arise when it is a proper subspace not of the form of span{(x1,x2,....,xm,0,0,...0)}. What I mean is it is not of the form of x,y, or z axis or xy,yz or xy plane as a subspace of R3.  It will not be closed with respect to multiplication. I think you should try to find a counter example having less modes compared to original dimension.

I hope you might have picked up the point from the vague argument I supplied.

Dear Rajesh,

Thank you for your comments. I understand you point. However, i am considering the

structure as a set of modes where each mode is defined by a frequency, damping ratio and modeshape. I then would like to define the addition operator as modal superposition for instance, or something like that. Likewise for multiplication using a proper definition of the multiplication of two modes..

Thanks for any hints. Regards, Maher