## 3 Mathematical Description
The orthogonal matrices \(\mathbf{Y}\) and \(\mathbf{X}\) as well as matrix \(\mathbf{N}\) are related to eigenvectors and eigenvalues of the following problems:  
\[\Big{(}\mathbf{B}\mathbf{B}^{\mathsf{T}}-\mu\mathbf{I}\Big{)}\mathbf{y}= \mathbf{0}, \tag{10}\]  
\[\Big{(}\mathbf{B}^{\mathsf{T}}\mathbf{B}-\lambda\mathbf{I}\Big{)}\mathbf{x}= \mathbf{0}. \tag{11}\]  
The solutions to the (10) and (11) eigen-problems, can be expressed as:  
\[\mathbf{M}=diag\Big{[} \begin{array}{cccc}\mu_{1}&\mu_{2}&\ldots&\mu_{n}\\ \end{array} \Big{]},\ \mathbf{Y}=\Big{[} \begin{array}{cccc}\mathbf{y}_{1}& \mathbf{y}_{2}&\ldots&\mathbf{y}_{n}\\ \end{array} \Big{]}, \tag{12}\]  
\[\mathbf{L}=diag\Big{[} \begin{array}{cccc}\lambda_{1}&\lambda_{2}&\ldots& \lambda_{m}\\ \end{array} \Big{]},\ \mathbf{X}=\Big{[} \begin{array}{cccc}\mathbf{x}_{1}& \mathbf{x}_{2}&\ldots&\mathbf{x}_{m}\\ \end{array} \Big{]}. \tag{13}\]  
The eigenvalues matrices \(\mathbf{M}(\in\mathbb{R}^{n\times n})\) and \(\mathbf{L}(\in\mathbb{R}^{m\times m})\) are dependent on the matrix \(\mathbf{N}\) of the decomposition (9):  
\[\mathbf{M}=\mathbf{N}\mathbf{N}^{\mathsf{T}},\ \mathbf{L}=\mathbf{N}^{\mathsf{T}} \mathbf{N}. \tag{14}\]  
One can notice that the product \(\mathbf{B}\mathbf{B}^{\mathsf{T}}\) can be considered as a matrix of symmetrized equations of equilibrium (5) with non-negative eigenvalues. Zero eigenvalues (if any) are related to the non-zero solution of homogeneous equations called a self-stress, or more precisely, self-equilibrated normal forces that satisfies homogeneous equations of equilibrium. The self-stress can be considered an eigenvector related to zero eigenvalue. In a similar way, the product \(\mathbf{B}^{\mathsf{T}}\mathbf{B}\) can be considered as a particular form of a linear stiffness matrix (7) with unit elasticity matrix \(\mathbf{E}\equiv\mathbf{I}\). The eigenvalues are non-negative. Zero eigenvalues (if any) are related to the finite or infinitesimal mechanisms, which can be considered an eigenvector related to zero eigenvalue. Finite mechanisms (rigid-body motions) refer to the motions that do not change the distance between any pair of nodes. Infinitesimal mechanisms describe the local geometrical variability in the range of small displacements. Generally, the information from the null-space analysis alone is insufficient to determine the difference. To establish whether the mechanism is infinitesimal or finite, it is necessary to apply nonlinear analysis with the use of geometric stiffness matrix (8), which is possible if the self-stress exists. If the eigenvalues of \((\mathbf{K}_{L}+\mathbf{K}_{G})\) are positive numbers, the mechanism is infinitesimal. It means the structure is stable. Zero eigenvalues are related to finite mechanisms. A negative eigenvalue represents instability of the structure. Full solutions to the eigen-problem of the stiffness matrix, taking into account the geometric stiffness matrix:  
\[(\mathbf{K}_{L}+\mathbf{K}_{G}-\sigma\mathbf{I})\mathbf{z}=0 \tag{15}\]  
can be expressed in the condensed form:  
\[\mathbf{K}_{L}+\mathbf{K}_{G}=\mathbf{Z}\mathbf{O}\mathbf{Z}^{\mathsf{T}}, \tag{16}\]  
where:  
\[\mathbf{O}=diag\Big{[} \begin{array}{cccc}\sigma_{1}&\sigma_{2}&\ldots& \sigma_{n}\\ \end{array} \Big{]},\ \mathbf{Z}=\Big{[} \begin{array}{cccc}\mathbf{z}_{1}&\mathbf{ z}_{2}&\ldots&\mathbf{z}_{n}\\ \end{array} \Big{]} \tag{17}\]  
are the matrices of eigenvalues and of normalized eigenvectors, respectively.