**4.1. Information produced by the Cayley–Hamilton theorem**

We show here that given *n* distinct eigenvalues for an *n*th order system, the Cayley–Hamilton theorem can produce at most *n*-independent equations, even though *n*<sup>2</sup> equations are produced through the entry-by-entry analysis. Let *p*(*t*) = *cnt <sup>n</sup>* + *cn* <sup>−</sup> 1*t <sup>n</sup>* <sup>−</sup> 1 + … *c*1*t* + *c*0 be the characteristic polynomial of an *n* × *n* matrix *A*. The Cayley–Hamilton theorem states that *p*(*A*) = *cnAn* + *cn* − 1*An* − 1 + … *c*1*A* + *c*0 is the zero matrix, which gives *n*<sup>2</sup> equations. Now, suppose that the matrix *A* is diagonal and let the diagonal entries be *λ*1, *λ*2, …, *λn*. The characteristic polynomial is

$$p\left(t\right) = \left(t - \lambda\_1\right)\left(t - \lambda\_2\right)...\left(t - \lambda\_n\right) \tag{10}$$

In this case, since *A* is diagonal, *p*(*A*) is also a diagonal matrix with exactly *n* equations. In fact, the *i*th diagonal entry is *p*(*λ<sup>i</sup>* ).

Now, consider the case that *A* is not diagonal. Given *n* distinct eigenvalues for *A*, then *A* is diagonalizable, so that *A* ' = *P*− 1*AP* is diagonal for some invertible matrix *P*. The characteristic polynomial of *A* ' is the same as the characteristic polynomial *p*(*t*) of *A*. In fact, the diagonal matrix *A* ' is the trivial solution to our design problem, and the 'structure' of the problem is housed in the matrix *P*. It is known that [15]

$$p\left(A\right) = P\left.p\left(A^\*\right)P^{-1}\right.\tag{11}$$

Equation (11) states that *p*(*A*) can be obtained by combining the equations contained in *p*(*A* '). At the same time, since *A* ' is itself diagonal, then matrix *p*(*A* ') contains exactly *n* equations. Thus, this states that *p*(*A*) is a combination of exactly *n*-independent equations and we can expect that although *p*(*A*) has *n*<sup>2</sup> equations, only *n* of them are independent.

It is also important to point out that the *n*-independent equations obtained from the Cayley– Hamilton theorem are each *n*th order polynomials. For a polynomial system with *n* unknowns and also *n* equations, then Bézout's theorem states that the problem has *nn* complex solutions [16].

#### **4.2. Required information**

It is clear that the spectral information (eigenvalues) alone is not enough to solve the problem. The method presented, along with all other methods, is limited by the fact that an *n*th order system can produce at most *n*-independent equations, even though *n*<sup>2</sup> equations are obtained 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 solutions at their disposal for creating suitable and optimized designs.

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 information is required to solve the problem.
