## Continuous dependence of eigenvalues

As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries.

Specifically, let be the complex vector space of matrices with entries in , endowed with any norm of our liking. As all norms are equivalent by the finite dimensionality of , we may for definiteness pick

(1)

(The fact that this norm is not sub-multiplicative is not relevant.) We recall that is the metric space of compact non-empty subsets of with the Hausdorff distance . The purpose of the present post is to prove continuity of the map that sends to its set of eigenvalues . For this it is sufficient to verify continuity of the map that associates with each matrix its characteristic polynomial. Once this is done, continuity of follows.

Now, it is known that the coefficients of the characteristic polynomial

can be expressed as sums of principal minors of , see e.g. of C.D. Meyer’s beautiful book Matrix Analysis and Applied Linear Algebra (SIAM, 2001). Clearly each principal minor of depends continuously on the entries of and therefore on itself. (The latter is most easily seen from (1).) Hence the same is true for the coordinate vector and, at last, for the characteristic polynomial .

## Continuous dependence of polynomial roots

In an interesting forum topic it was recently asked to show that the roots of a real or complex polynomial depend continuously on the polynomial’s coefficients. Although I have used this proposition numerous times, implicitly and explicitly, I realized that I never saw a proof of it.

Perhaps the most obvious approach would try to apply the Implicit Function Theorem but, as you may know or can easily check, such an attempt would only work for roots that are simple. Indeed, the very failure of the Implicit Function Theorem in case of non-simple roots is one of the subjects studied in local bifurcation theory. For an example, see this discussion of the Bogdanov-Takens bifurcation.

Returning to the original proposition, here is an elementary proof using the Fundamental Theorem of Algebra.

##### Preliminaries
Let be the set of all polynomials of degree at most with complex coefficients. For any let be the coordinate vector of with respect to the standard basis . If is the maximum norm on then

turns into a normed space. For with and their pointwise difference satisfies

(1)

for all , where . Define as the open subset of complex polynomials of degree precisely and let be the collection of non-empty, compact subsets of . When endowed with the Hausdorff metric this collection becomes a metric space. Define to be the map that assigns to its (non-empty and finite) set of roots. In these terms we have

(2)

where and are the usual point-set distances in the complex plane from to and to , respectively.

##### Estimating the first term inside
We show that is continuous at any point . Let be given. The two terms appearing inside the braces in (2) will be estimated separately. By the Fundamental Theorem of Algebra every with leading coefficient decomposes as

where are the roots of , repeated according to multiplicity. So, when it follows that

(3)

where the inequality is due to (1). Now,

so and therefore (3) implies that

(4)

Hence there exists such that implies that the left-hand side of (4) does not exceed and thus for at least one . Consequently,

whenever . We set . This takes care of the first term inside the braces in (2).

##### Bounding the roots by the coefficients
Suppose is a root of with coordinate vector . Since it is immediate that

provided that we assume . This yields the bound

(5)

of the roots of a polynomial in terms of its coefficients.

##### Estimating the second term inside
Applying the Fundamental Theorem of Algebra once more, we write

where this time are the roots of . Let and let . Then

(6)

By (5) the coefficient is bounded on every sufficiently small ball in centered at . Hence there exists such that implies that the left-hand side of (6) does not exceed and therefore for at least one . It follows that

provided .

We finish by puting . Then (2) shows that whenever .

## Laatste-wil-pil

Gisterenavond las ik op NU.nl en de website van dagblad Trouw dat het voor senioren mogelijk moet worden een eind aan hun leven te maken door inname van een “laatste-wil-pil”. Aldus is althans het streven van de Nederlandse Vereniging voor een Vrijwillig Levenseinde en de Coöperatie Laatste Wil, die hiervoor steun vinden bij Tweede Kamerlid Pia Dijkstra (D66). Zij schrijft momenteel aan een wetsvoorstel van dergelijke strekking. De doelgroep bestaat nadrukkelijk uit ouderen die volgens de huidige euthanasiewet niet in aanmerking komen voor hulp bij zelfdoding, maar wel hun leven willen beëindigen omdat “ze er zelf helemaal klaar mee zijn en er echt niets meer van kunnen verwachten.”

Mij werd het bij het lezen van dit nieuwsbericht zwaar te moede. Ik moest al snel denken aan opgeluchte (klein)kinderen die zich binnenkort verlost weten van verplichte zondagmiddagbezoekjes aan een muf bejaardentehuis en aan handenwrijvende voorzitters van onderpresterende pensioenfondsen.  Bij het ministerie van Volksgezondheid vermoed ik eveneens een zeker enthousiasme. Ik verbaasde me dan ook niet over de toezegging van de VVD om te zijner tijd “welwillend” naar Dijkstra’s voorstel te zullen kijken.

Maakt “de pil” straks deel uit van menig geschenkje voor de kersverse tachtiger? Wordt in het schriftelijke pensioenoverzicht van elke vijfenzeventigjarige binnenkort een subtiele suggestie opgenomen?

Nee, zo een vaart zal het vast niet lopen. Veel storender dan bovenstaande fantasieën vind ik de boodschap die wordt afgegeven. Mocht het ooit tot een laatste-wil-pil komen, dan wordt daarmee in onze samenleving de dood een legitiem antwoord op problemen van niet-medische aard. Tegelijkertijd houden zowel de NVVE als ook Dijkstra vast aan toetsing door een arts, iets dat mij voorkomt als tweeslachtigheid. De Coöperatie Laatste Wil heeft van een dergelijke inconsequentie trouwens geen last: Zij stelt de pil graag aan iedere meerderjarige ter beschikking, zonder tussenkomst van een medicus.

Zelfdoding is een moeilijk onderwerp en algemene uitspraken doen hier dan ook zelden recht aan het lijdende individu. Nochtans gaat het idee van een laatste-wil-pil mij meerdere bruggen te ver. Als uitgesproken tegenstander van vrijwillige euthanasie op chronisch geesteszieken (de doodswens maakt hier immers vaak deel uit van het ziektebeeld)  gruw ik helemaal van de gedachte dat de overheid actief dan wel passief (bijvoorbeeld door het afzien van strafvervolging)  hulp bij niet medisch geïndiceerde zelfdoding faciliteert.

Hij die lijdt verdient geen pil, noch veroordeling, maar juist aandacht en steun om zijn leven, in welke fase dan ook, weer zijn moeite waard te maken.

## Why I started this weblog

Since September I have been a member of a well-known internet forum on science, mathematics and engineering. Overall, this was a very enjoyable and stimulating experience. At some point I made it to “science adviser”, which I believe is a term forum staff reserves for members that display a better than average understanding of the topic at hand in technical discussions.

The forum also has an active non-technical section for the discussion of current events, and this is where my experience eventually turned sour. On the 17th of February in a thread on the recent death of Antonin Scalia, former member of the supreme court of the USA, I posted a link to this blog entry by Mrs. Debra Loevy, a civil rights lawyer from Chicago. In her blog Mrs. Loevy exemplifies Scalia’s questionable legacy by reminding us of the disgraceful case of Troy Davis, who was executed by the state of Georgia in 2011 in spite of serious doubts about his guilt.

Already the same day, the forum’s moderators claimed that Mrs. Loevy’s blog contains “false or misleading information” and they promptly deleted the link from the Scalia thread. Upon my further inquiry Mrs. Loevy’s blog post on Scalia was harshly criticized and ultimately qualified as “disingenuous”.

I decided to contact Mrs. Loevy directly and she was willing to review the moderators’ remarks. Her written reply, received by the forum staff no later than March 4th, was both comprehensive as well as factual. Unfortunately, notwithstanding the forum administrator’s cooperation, to this date none of the moderators involved could be bothered to react in any way.

This outcome is not only a sign of the weakness of the moderators’ position. It has also strengthened my belief that their decision to erase the link to Mrs. Loevy’s blog was not motivated by an interest in facts, but rather by a desire to favor their own views while suppressing unfavorable assessments of Antonin Scalia’s legacy. This practice reeks of abuse of power and is detrimental to the quality and progress of any discussion, academic or otherwise.

In conclusion, I would like to thank Greg Bernhardt for my pleasant stay at his forum, now that I have decided I will not resume my contributions there. Instead, I plan to post on the present weblog my thoughts on mathematical problems of varying degree of difficulty, as well as my comments on other topics of my choice. Perhaps someone will read my posts, perhaps not.

Finally, my gratitude goes to Mrs. Debra Loevy for her friendly help and interest. If you are curious about a lawyer’s perspective on certain social aspects of the American justice system, you are encouraged to visit her firm’s blog.

Update (August 2016): Following some additional conversation with Greg, I have decided to give Physics Forums another try. Let us see how things will go.