Answering, some time ago, to this question : Change in eigenvalues on changing one entry of a matrix, I had the idea of a graphical representation of roots of polynomial equations $P(x,a)=P_{a}(x) \in \mathbb{R}[a,x]=\mathbb{R}[a][x]$, such as

$$x^3+a x^2+(2a+1)x-3a=0.$$

In fact, I will use notation $\lambda:=x$, because we can assume WLOG that such polynomials can always be considered as characteristic polynomials of certain matrices whose coefficients are polynomials in $a$ (companion matrix for example).

I propose a graphical simultaneous representation of real and complex roots $\lambda_1,\lambda_2 ... \lambda_n$ of $P_a(\lambda)=0$ displaying their global dependency on real parameter $a$ (see Fig. 2).

This representation will be explained through an example [with a certain similarity with the example given in the reference] in order to convey easily the underlying ideas.

Let us consider the characteristic polynomial of the following matrix with one of its entries considered as a variable, here the bottom left entry (it is a particular case of what is called a pencil of matrices) :

$$A=\left(\begin{array}{rrrr}1& 3& 0& -2\\ -1& -2& 0& 3\\ -1& 2& -2& -2\\ \color{red}{a} & 3 & 0 & 0 \end{array}\right).$$

A plot of the real roots of $\det(A-\lambda I)=0$ is given in Fig. 1 : for a given value of $a$, we plot points $(a, \lambda_k)$. We see in this way regions with two or four real roots, exceptionnaly 3. The eye is attracted by the "interweaving curves" thus generated that will be analyzed later on.

enter image description here

Fig. 1.

The "3 roots" case corresponds to places of bifurcation (little red circles) where 2 formerly distinct real roots coalesce and then "disappear", or, in a reverse way, appear "out of nothing".

Let us proceed by showing how these appearances/disappearances can be made visible (revisiting the well known fact that coalescence of real roots into a pair of complex conjugate roots, and vice versa, is a well-known phenomena).

Take a look at (3D) Fig. 2, and compare it with (2D) Fig. 1. Instead of having a real-valued axis for $\lambda$, we switch to a "complex valued axis", more precisely, we replace a single axis by two of them, one for the real part, the other one for the imaginary part. As complex eigenvalues come by conjugate pairs (because the polynomials we consider have real coefficients), Fig. 2 is symmetrical with respect to the horizontal plane defined by $a$ and $\Re(\lambda)$ axes.

In this way, we can "track" the roots and understand in a finer way what happens for certain values of the parameter.

Our understanding of Fig. 1 can be refined : see Appendix below.

enter image description here

Fig. 2. "Complexification" of Fig. 1. In order to gain a better 3D perception, here is a chinese poetic interpretation : imagine the blue lines as borders of a lake connected by a red bridge with a leaning weeping willow trunk and their magenta reflections on the lake...

Here is a second example (Fig. 3), associated with matrix :

$$A=\left(\begin{array}{rrrr}0& 1& 0& \color{red}{-a}\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ \color{red}{a} & 0 & 1 & 0 \end{array}\right).$$

enter image description here

Fig. 3.

I have willingly taken simple examples with at most first degree entries in parameter $a$ in order to demonstrate the interest of this approach. I have tried different cases, some of them with higher degrees. Representations analogous to Fig. 2 and Fig. 3 display, rather often, features with heuristical/pedagogical interest.

Question : This representation is very likely to be found elsewhere and/or linked to a certain already developed theory (differential geometry, Galois theory...). Can somebody give me an answer or at least a track ?

Remark : this representation could be considered as trivial as its "slices" at $a$= constant are just copies of the roots' position in the complex plane (see the comment of @Andrea Marino), a kind of continuous juke-box...

Edit: Meanwhile, I have found some references:

  • "Complex Bifurcation from Real Paths" by M. E. Henderson and H. B. Keller, SIAM J. Appl. Mat. Vol. 50, No. 2, pp. 460-482, April 1990, that can be uploaded here.

  • And this special approach.

  • I mention also "Eigenvalue attraction" by R. Movassagh (https://arxiv.org/pdf/1404.4113.pdf), with a different point of view.

Appendix : A deeper understanding of Fig. 1:

The two "complex domain curves" (red, and magenta for their complex conjugate parts) are

  • on the right, a parabolic arc, thus planar, whereas,

  • on the left, the ellipse-looking curve is not planar.

Consider now the (blue) real components (the straight line, and the two curves of the horizontal plane); let us play a game of anticipation:

  • for the straight line, we guess the presence of a root $\lambda=-2.$

  • for the two curves, it is normal to consider them together as having the same equation with a horizontal common asymptote for a certain value (of course never reached) of $\lambda$. Zooming on the graphics, one can guess that this asymptote is characterized by $\lambda_{\infty}=2.5$, Besides, a little observation makes evident a parabolic envelope ; packing these observation together, a possible equation for $a$ as a function of $\lambda$ is as follows :

$$\tag{1}a = p\lambda^2+ q\lambda +r+\dfrac{s}{\lambda-2.5}.$$

for certain coefficients $p,q,r,s$. It remains to check (1) by examining the factorization of the characteristic polynomial of $A$ and then its roots. Using a CAS, or by hand calculation, we obtain:

$$\tag{2}\det(A-\lambda I)=(\lambda+2)P(\lambda) \ \ \ \ \text{with} \ \ \ \ P(\lambda):=\lambda^3+\lambda^2-8\lambda+2a \lambda+3-5a$$

Equating (2) to zero, we obtain $\lambda=-2$, as awaited, or $P(\lambda)=0$, which can be written in the following equivalent way :

$$a=\dfrac{-\lambda^3-\lambda^2+8\lambda-3}{2 \lambda -5}$$

in full conformity with form (1).

  • 1
    $\begingroup$ What you described is a family of varieties, I think. Precisely, you have the set of solutions $V$ of a polynomial $f$ in two variables $(a, \lambda) $ which you depicted for real $a$ and complex $\lambda$. You have a projection $\pi: V \to \mathbb{R} $ such that $\pi(a, \lambda) =a$. Fibers of this map $V_a=\pi^{-1}(a)$ are the solutions with fixed $a$: that's why this is called a family, parameterized by a goemetric space $\mathbb{R} $. $\endgroup$ – Andrea Marino Jan 19 '18 at 0:11
  • $\begingroup$ @Andrea Marino Very interesting. Thanks. $\endgroup$ – Jean Marie Jan 19 '18 at 5:23
  • $\begingroup$ @Andrea Marino I just found a reference that I have given at the bottom of my text, in compliance with your remark. $\endgroup$ – Jean Marie Jan 19 '18 at 7:11

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