**4.4. Partially described systems**

through applying the generalized Cayley–Hamilton theorem, as discussed above. If the matrices are completely unknown, then there may be as many as 2*n*<sup>2</sup> unknown entries in the *M* and *K* matrices. The Cayley–Hamilton method can produce at most *n*-independent variables and the remaining equations must be specified in other ways. For creating physically realistic systems, this generally entails preconditioning or constraining the structure of the matrices.

For solving discrete conservative vibration problems, Equation (6) can be used as long as the *M* matrix is non-singular, and equation (7) can be used if *M* and *K* commute. It becomes very clear by considering the example in equation (9) that more information is required in order to build a suitable vibrating system based on *M* and *K*. From an engineer's point of view, any system, which can produce similar eigenvalues, has the potential of being a suitable solution, as long as it fits within the physical criteria set at the outset of the project. It is obvious that the example in Equation (9) can produce an infinite number of possible solutions. Equation (9) represents one of the several solutions to the Equation 8. As the system's order increases, so does the number of potential solutions (Bézout's theorem). Thus, engineers have many

The inverse problem is then not limited by the eigenvalues, and in fact the eigenvalues alone do not contain enough information from which to build a physical system. Therefore, other

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

> 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–

solutions at their disposal for creating suitable and optimized designs.

information is required to solve the problem.

reduce the number of unknowns from 2*n*<sup>2</sup>

**4.3. Structural constraints**

34 Applied Linear Algebra in Action

Another aspect that affects the amount of information required is the information available at the outset of the problem. Thus far, the entire spectral set has been used as input information, as well as specification of matrix entries when necessary. However, as stated in the study of Chu [1], often only portions of the entire spectrum are available. This is termed as a partially described inverse eigenvalue problem. This is true whether the subset of available spectral data arises from the design requirements or from the experimental data obtained for system identification. Regardless of the information available, the Cayley–Hamilton theorem can be used to produce *n* pieces of information for an *n*th order system stemming from *n* degrees of freedom. If certain eigenvalues are missing, the Cayley–Hamilton theorem can still be used.

Typically, in order to completely solve an *n*th order inverse problem using the Cayley– Hamilton theorem, only *n* unknown values should be present in the problem, regardless of the manner of their appearance along the solution path. Problem C makes full use of this detail during its solution by preconditioning the unknown matrix to account for this. For example, for a 3DOF system, it would be expected that only three unknowns should be present. Consider for example, the following system that has three known eigenvalues *λ*1 = 0.47, *λ*2 = 4.66, *λ*3 = 10.87 and three unknown entries in the matrix given by

$$A = \begin{bmatrix} a\_1 + a\_2 & -a\_2 & 0 \\ -a\_2 & a\_2 + a\_3 & -a\_3 \\ 0 & -a\_3 & a\_3 \end{bmatrix} \tag{12}$$

Using the Cayley–Hamilton theorem as described in Section 3, the matrix can be determined such that *a*1 = 9.43, *a*2 = 1.23, *a*3 = 2.06, which gives a matrix solution for the system as

$$A = \begin{bmatrix} 10.66 & -1.23 & 0 \\ -1.23 & 3.28 & -2.06 \\ 0 & -2.06 & 2.06 \end{bmatrix} \tag{13}$$

As in Problem C, it is often the case that only partial spectral information is available. Therefore, the Cayley–Hamilton theorem can be used if the amount of missing spectral information is replaced by the same amount of matrix entry information. So, if only *λ*3 = 10.87 is known then two of the three *a* matrix entries must be known. If *a*2 = 1.23 and *a*3 = 2.06 are known in the example above, then solving the Cayley–Hamilton equations gives *a*1 = 9.43, *λ*1 = 0.47, *λ*2 = 4.66. Consequently, solving Problem C is not much different from solving Problem A. The same is true if the system is made up of a matrix *M* and a matrix *K*, except in this case, the generalized Cayley–Hamilton theorem must be used.

It becomes clear that one of the largest difficulties in using inverse eigenvalue theory to solve many physical problems is setting up the appropriate matrix form of the problem. This is achieved by consideration of the modelling approach chosen to model the forward problem. In most cases, the forward solution will utilize some form of discretization, such as finite elements, global elements, finite difference, and so on. The method chosen has a large impact on the structure of the matrix, making the solution easier or more difficult depending on the situation. Also, in discretizing, the method chosen to relate material parameters has a large effect on the number of independent variables. Therefore, it is extremely important to ensure that simplification methods are properly considered.
