We are now left with a quadratic equation (4.17). Initially this poses a problem because a quadratic equation has two solutions (or roots); in other words, X can have two values. But earlier we had assumed that the square matrix M could be replaced by a single value, X. Effectively this becomes true as the larges of the two X values, referred to as the dominant root, has most influence on the dynamics. Note that the dominant root may be complex or negative. A negative dominant root is biologically meaningless in this application (but see Chapter 7) whereas complex roots are discussed in Chapter 7. In mathematics the values of X are called the eigenvalues and the corresponding values of R and F are the eigenvectors. The eigenvalues may also be referred to as the latent roots or the characteristic values of the matrix, M. Similarly, the eigenvectors are known as the latent or characteristic vectors. (In passing it is worth noting that in finding values for R and F we have found solutions for the equations 4.9 and 4.10. Matrix methods have a wide application in the solving of simultaneous equations.) Finally, it may be helpful to know that equations 4.13-4.16 can be written in a general mathematical shorthand for any size of matrix M and vector v (as equation 4.8):
The requirement for the non-trivial solution is that |M - XI = 0
with values of X being found by solution of the characteristic equation.
To reinforce all these theoretical points let us consider a specific example. If f = 100, s01 = 0.1 and s12 = 0.5 then from equation 4.17:
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