**4.3. Structural constraints**

One of the easiest ways to limit the number of matrix entries and include structural constraints is by including zero entries or by incorporating symmetry into the matrices. These constraints are interesting because they follow directly from real systems. For example, for many simply connected spring-mass systems, only diagonal entries are present in the mass matrix, and the stiffness matrix is expected to be symmetric and therefore can be mathematically forced to be symmetric by imposing that symmetric off-diagonal terms be the same. Physically, the structure of these matrices reflects the fact that the degrees of freedom of the system are coupled through stiffness element couplings only and no mass coupling is present. Furthermore, the symmetry of the stiffness matrix is a consequence of Newton's third law. The system is further constrained by the fact that the off-diagonal terms in the stiffness matrix are not independent, but are related to the diagonal terms, once again reducing the number of unknowns. In this case, incorporating physical constraints into the mathematical structure of the problem can reduce the number of unknowns from 2*n*<sup>2</sup> to *n*<sup>2</sup> .

In this case, the engineer still has the freedom to choose from several systems, which would satisfy the requirements. Further constraints may come in the form of available components, such as stiffeners, which must fit within specified physical dimensions or the financial budget.

These observations highlight the importance of the forward problem. Continuous systems are typically discretized using various methods, which produce a model of the system in matrix form. The form of the matrix is heavily dependent on the choice of discretization method. The inverse problem is consequently affected because it seeks to create a matrix that matches the form as chosen at the outset by the choice of discretization approach. Although the Cayley– Hamilton theorem does not discriminate in its ability to solve these various matrix forms, it is possible that certain discretization methods lead to simpler forms or matrices containing fewer variables. In this sense, an inverse problem may be easier to fully define.
