**4. Analysis**

Increasing the size of the square matrices increases the number of independent variables faster than it does the dependent variables. In other words, for an *n*th order system with *n*-specified

in this chapter, the generalized Cayley–Hamilton theorem only produces *n*-independent

In order to use the Cayley–Hamilton theorem as a tool to find the *n* dimensional generalized eigenvalue system problem based on knowledge of desired eigenvalues and/or physical parameters, 2*n*<sup>2</sup> − *n additional* pieces of design information are required in addition to the *n*-

ALGORITHM. *Given the n*-*desired eigenvalues λ*1, …, *λ<sup>n</sup> of an nth order generalized eigenvalue*

( ) ( ) ( )( ) <sup>1</sup>

 ll

**2.** *Generate forms for the M and K matrices from physical parameters and using the additional* 2*n*<sup>2</sup> − *n pieces of design information*, *applying symmetry and any other techniques based on the*

( ) () () () <sup>1</sup> 11 1

( ) 1 1

**5.** *Select n non*-*zero independent equations* (*the Cayley*–*Hamilton matrix diagonal entries work well*);

**8.** *Verify that a valid solution is obtained by calculating* det(*λM* + *K*) = 0 *and ensuring that the*

*equations from the Cayley*–*Hamilton equation in step 3*;

**7.** *Insert the n computed values into their appropriate places in the M and K matrices*;

**6.** *Compute the n unknowns by solving the n*-*independent equations*;

*The output consists of the n matrix entries of the M and K matrices*.

*n nn n p*

 ll

*discretized forward problem and leaving an n number of unknowns*;

1 1 1 10

*c c cc* - = - ¼ - - = + +¼+ + - -

1 10 , 0 *n n n n p KM c M K c M K c M K cI* - -- - = + - ¼+ + =

1 10 , 0 *nn n n n p K M c K c K M c KM c M* - - = + ¼+ + = -

 l

*n n*

 l  l

desired eigenvalues. An algorithm for solving Problem B is as follows:

**1.** *Generate the characteristic polynomial*:

l ll

**3.** *Generate the Cayley*–*Hamilton theorem equation*:

*or for commuting M and K matrices*:

*initially desired eigenvalues are obtained*.

**4.** *Extract all n*<sup>2</sup>

unknown variables are required to find *K* and *M*, but, as will be shown later

eigenvalues, 2*n*<sup>2</sup>

32 Applied Linear Algebra in Action

**3.4. Algorithm**

equations.

*system*

In this section, results on the number of independent equations and required pieces of information used in the prior sections are discussed.
