What do eigenvectors physically represent in vibrating systems?

[Alexander79]

I'm an engineering student working on a vibration analysis project, and I'm using eigenvalues from a system's mass and stiffness matrices to determine its natural frequencies. While I can perform the calculations, my conceptual understanding feels shaky; I'm struggling to intuitively grasp what the eigenvectors physically represent beyond just being modes of vibration. For those who use eigenvalues in applied settings, how do you visualize or explain the physical significance of an eigenvector in a real-world system, and are there any tricks for building that intuition beyond the abstract mathematical definition?