In a May 4, 2012, post to Maple Primes, gdorsch sought the locus of eigenvalues for parameter-dependent matrices. In reply, Markiyan Hirnyk proposed the matrix  as a model to study how the eigenvalues of the matrix  morph to the eigenvalues of  as the parameter  varies from 0 to 1. Apparently, the matrices  and  are real.

If the matrices  and  are not symmetric, the eigenvalues of  can become complex, making the analysis of their loci very difficult. However, on May 30, 2013, jschulzb appended to the earlier post the comment "I have the same problem. Have you found a good solution yet?" and the post branched to a separate new question about eigenvalue ordering. In this new post, jschulzb imposed symmetry on the matrices  and , making the problem slightly more tractable.

This question about the loci of eigenvalues is reminiscent of two earlier investigations in [1] and [2]. In [1], the general question of the root locus is raised. In the context of feedback control, engineers use the locus of roots of certain polynomials to determine the stability of these feedback systems. The characteristic polynomials for the matrices  certainly raise similar issues.

In [2], Mike Monagan presents a Maple solution to an exercise in [3], a linear algebra text that touts numeric software as the best tool for studying that subject. In this text, Steve Leon essentially asks the student to explore the transition of the eigenvalues of one matrix to the eigenvalues of another. Because of the limited availability of access to back issues of the MapleTech journal, the next section of this article will be a paraphrase of Monagan's "reply" to Steve Leon.

Thereafter, this article will explore several examples of the loci generated by symmetric real n × n matrices , for , and 5. These examples will show that whether or not the eigenvalues are degenerate for some , loci of eigenvalues have to be as taken curves with continuously turning tangents, that is, as  curves.

Given that the earlier editions of [3] are dated 1980, 1986, 1990, and 1994, it must be obvious that Mike Monagan's MapleTech article was based on one of the first three editions of Steve Leon's text. Mike's article used matrices that differed slightly from those in Exercise 4, page 360, in [3].

The following is a paraphrase of the exercise, and consequently, of the MapleTech article.

Solution

The eigenvalues of  are  whereas the eigenvalues of  are . Because  is not symmetric, its eigenvalues can be, and actually do become, complex. Indeed, the eigenvalues of  are . Figure 1 is a graph of the real values of these eigenvalues; Figure 2 shows the eigenvalues in the complex plane. In each figure, , with  drawn in black; and , in red.

Figures 3 and 4 repeat Figures 1 and 2, respectively, but for . Figure 3 makes it clear that the analytic expressions giving  are discontinuous as real functions, but not as complex-valued functions. On the other hand, Figure 4 shows that in the complex plane, the loci of  do not have continuously turning tangents. Although  will be symmetric in the remaining examples in this article, Figures 3 and 4 portend some of the difficulties that will be encountered even for a restricted class of matrices.

Hence, the loci of  bifurcate at  and . But the real question is, do the closed-form expressions  each define the locus of an eigenvalue (resulting in the black and red curves in Figure 4), or are the loci of the eigenvalues the connected curves in Figure 3?

Solution

Solution

Solution

Solution

The closed-form expressions for , are continuous, but not . (This can be established analytically by the calculations in Table 1.) So either the loci of the eigenvalues are defined by the closed-form expressions  and therefore do not have continuously turning tangents, or the loci are smooth curves and are only piecewise-defined by the analytic expressions  whose graphs appear in Figure 10. A precise definition of the locus of eigenvalues of a real, symmetric, matrix  is required.

Solution

Slopes on either side of  can be calculated numerically from  computed implicitly from the characteristic polynomial.

Recall that  in  must be replaced by the appropriate eigenvalue  which is available only through numeric calculation via the function . The limiting process used in Table 1 can't be applied here; the requisite numeric calculations are summarized in Table 2.

Solution

Since  is a 5 × 5 matrix, only numeric techniques can be used to find its eigenvalues. However, note that the matrix has been chosen so that the eigenvalues of  are 1, 5, 10, 10, and 15.

In Table 3 the matrix  is defined, and the characteristic polynomial is defined as the function .

The characteristic equation, (here, ) defines the loci of the eigenvalues of  implicitly. Figure 12 implements this insight via Maple's implicitplot command applied to the characteristic equation.

Table 4 shows the calculations needed to solve for the eigenvalues numerically, and to construct the separate loci based on these numeric data.

Figure 13 assembles the graphs , into a single graph via Maple's display command; it shows that the green and blue curves share the common point .

For the green and blue curves in Figure 13, slopes on either side of  can be calculated numerically from  computed implicitly from the characteristic polynomial. Table 5 contains the relevant calculations.

There are no closed-form expressions for the eigenvalues of this 5 × 5 matrix . The eigenvalues are computed numerically by Maple's Eigenvalues or fsolve commands, each of which return a sorted list of eigenvalues. There is no user-control of this sort, but even if there were, what sorting rule could be invoked across an eigenvalue with algebraic multiplicity greater than 1? It would seem that the only way to define a unique locus of eigenvalues is to require that it be of class , that is, that it have a continuously turning tangent.