I am an engineer working on a structural engineering analysis problem. It is quite simple: I have a 3x3 Matrix representing the stiffness of a 2D-structure (with the three degrees of freedom: displacement in x-direction $u_x$, displacement in y-direction $u_y$ and rotation about z -axis $\varphi_z$).
The Matrix $K$ represents the Stiffness. With a given force vector $F=\{{f_x;f_y;\varphi_z}\}$. The displacement vector $U$ is calculated from $U=K^{-1}\cdot F$.
Now in some cases $K$ is not invertible, and I would like to find the degree of freedom, that is responsible for that. The structure may be kinematic in one direction or the rotation. I know, that K is not invertible, if $det[K]=0$ or near $0$. But how do I calculate the eigenform using Singular Value Decomposition?
An example:
Thank You and sorry for my poor mathematic knowledge.