**6. The determinant method for partially described systems**

Although effective, the Cayley–Hamilton method for solving inverse eigenvalue problems described in the previous section may not be the most computationally efficient method when solving partially described systems. Partially described systems are those for which only a select few eigenvalues are specified, as opposed to the entire spectrum being given. In this section, the determinant method, which first appeared in the study of Mir Hosseini and Baddour [21], is introduced.

For clarity, the method is explained for the case where two eigenvalues are required to be a certain value. The method proceeds in the same manner where the number of specified eigenvalues is larger. Solving the characteristic equation of the system requires the use of the determinant equation

$$p\left(t\right) = \det\left(K - \lambda M\right) = 0\tag{18}$$

where *λ* is the eigenvalue obtained by finding the roots of the polynomial equation 18. Suppose that two of the possible *n* roots (in this case the eigenvalues *a1* and *a2*) are required to be of a certain value. Replacing *λ* in the characteristic polynomial with the desired *a1* and *a2* each in turn leads to two scalar equations, specifically

$$\begin{aligned} p\left(a\_1\right) &= \det\left(K - a\_1M\right) = 0\\ p\left(a\_2\right) &= \det\left(K - a\_2M\right) = 0 \end{aligned} \tag{19}$$

The two scalar equations in equation (19) can be simultaneously solved for up to two un‐ knowns. This greatly reduces the computational effort involved. Constraints can be used to create a physically realistic system. Similar to the case for the Cayley–Hamilton theorem, the equations will be polynomials and the use of a good polynomial solver is required to obtain the multiple solutions expected.

#### **6.1. Algorithm**

In the case of Problem C, *m* eigenvalues are given for an *n* dimensional system, so that *n* − *m* is the number of unknown or unspecified eigenvalues for the system. The output of the determinant method consists of the remaining *n* − *m* eigenvalues as well as any of the unknown matrix entries.

Determinant Algorithm. *A certain set of desired eigenvalues*, *a*1, …, *am*, *which is smaller than the dimension of the system*, *are given as input information*.


*p*(*a*1) = det(*K* − *a*1*M*) *and p*(*a*2) = det(*K* − *a*2*M*)…*p*(*am*) = det(*K* − *amM*)*where a*1, …, *am are obtained from the desired natural frequencies*;

