Análisis inferencial

This section gives weak sufficient criteria for the existence of fixed points, simply by ex- ploiting the contraction mapping theorem and Brouwer’s fixed point theorem. The exis- tence theorems here are useful in the case of (entrywise) positive matrices, but in general are a faint shadow of what the final results ought to be.

Letᏸbe an element of End MnC. Define the following generalization of spectral radius

forᏸ. For each norm · on MnC, define the corresponding operator norm||| · |||in

the usual way, that is,|||ᏸ||| =sup_Y=000ᏸ(Y)/Y. Now defineρn(ᏸ)=inf{|||ᏸ||| | · is an algebra norm on MnC}. Recall that analgebra norm(in this case on MnC) is a

Banach space norm with the additional property thatY Z ≤ Y · Zfor all pairs of elementsY andZ. We can always renormalize so that I =1, and so incorporate this into the definition.

An easy consequence of Jordan normal form is that for anyk×kmatrixA,

ρ(A)=infA | · is an algebra norm on MnC

. (14.1)

Applying this to End MnC∼=Mn2C, we see that all the norms on the latter arising in

the display above are algebra norms, henceρ(ᏸ)≤ρn(ᏸ). Nowρ(ᏹC,D)=ρ(C)·ρ(D)

(use the natural tensor product decomposition), and so it is easy to calculate the former. Unfortunately, for proving results on the existence of fixed points, it isρn(ᏹC,D) that

matters, and in many cases,ρn(ᏹC,D)> ρ(C)·ρ(D).

For a square matrixC,C∗will denote conjugate transpose.

Proposition14.1. ForCinMnC,ρn(ᏹC,C∗)=ρ(CC∗).

Proof. SinceᏹC,C∗(I)=CC∗ and for every algebra norm on M_nC we have CC∗ ≥ ρ(CC∗), it follows thatρn(ᏹC,C∗)≥ρ(CC∗). On the other hand, for any algebra norm, CY C∗ _{≤ C · C}∗ _{· Y, whenceρ}

n(ᏹC,C∗)≤inf{C · C∗}, where the norm

varies over all algebra norms. If · 2denotes the usual (2-2) operator norm on MnC

(acting onl2_{(1, 2,. . .,n) in the usual way), we haveρ}

n(ᏹC,C∗)≤ C2· C∗2= CC∗2

=ρ(CC∗).

If, in the context of this result,Cis not normal, thenρ(CC∗) can be strictly bigger thanρ(C)2_{. For example, this occurs ifCis nilpotent, or consists of Jordan blocks with} at least one being of size exceeding one, and corresponding to an eigenvalue of maxi- mal modulus. These yield (by small perturbations) examples whereChas only strictly positive entries. An obvious inequality is thatρn(ᏹC,D)≤inf{C · D}(restricted to

normalized algebra norms).

On the other hand, whenCandDare both normal (but not necessarily related to each other in any way),ρn(ᏹC,D)=ρ(C)·ρ(D). In fact,ᏹC,Dsatisfies a stronger property. For

ᏸin End MnC, we sayᏸachievesρn(ᏸ) if there is an algebra norm · on MnCsuch

thatρn(ᏸ)= |||ᏸ|||where||| · |||is the operator norm induced by · .

Proposition14.2. IfCandDare normal matrices of sizen, thenρn(ᏹC,D)=ρ(C)·ρ(D) and moreover, this is achieved by the 2-2 norm onMnC.

Proof. Obviouslyρn(ᏹC,D)≤ C2D2, and since the matrices are normal, this equals

ρ(C)·ρ(D). Hence equality occurs.

Lemma14.3. IfC=Ior ifD=I, or ifCD=DC and one ofC,Dis diagonalizable with

distinct eigenvalues, thenρn(ᏹC,D)=ρ(C)ρ(D). In the last case,ᏹC,D achievesρn. In the former cases,ᏹC,D achieves ρn if and only if the nonidentity matrix of the pair has the property that for every eigenvalue of modulus equaling the spectral radius, all corresponding Jordan blocks are of size one.

Proof. If one of the pair is the identity, all the results aboutᏹC,Dare routine and follow

from the earlier observation about algebra norms on MnC. IfCD=DCand (say)Cis

diagonable with distinct eigenvalues, then diagonalizingCautomatically diagonalizesD

(since the centralizer of a diagonal matrix with distinct eigenvalues consists of diagonal matrices). IfA does the diagonalizing, take the normA·A−1

observe thatρnis invariant underᏸ→ᏹA,AᏸᏹA−1_,A−1, and applying this toᏹ_C,Dyields

diagonal, hence normal, matrices, so the preceding applies).

The functionρnis introduced in order to use the contraction mapping and Brouwer’s

fixed point theorems.

Proposition14.4. Suppose thatCandDare square matrices of sizen. Letφ:X→(I−

CXD)−1_{be the corresponding fractional linear transformation.}

(a)Ifρn(ᏹC,D)<1/4, thenφis contractive on the 2-ball ofMnCfor some algebra norm, and there exists an attractive fixed point of norm less than2. Moreover,{φN_{(000)}converges} to the fixed point.

(b)Ifρn(ᏹC,D)=1/4andᏹC,Dachievesρn, thenφhas a fixed point of spectral radius at most2.

Proof. In the first case, there exists an algebra norm · on MnCsuch that|||ᏹC,D|||<

1/4 in the corresponding operator norm, and thus for everyYin the 2-ball (using · ), we haveᏹC,D(Y)<Y/4≤1/2. Hence(I−CY D)−1<1/(1−1/2)=2. Soφacts as a (strict) contraction on the 2-ball, and thus by the contraction theorem, all results in (a) follow.

In the second case, using the norm that achievesρn, the same calculation as in (a)

yields that the closed 2-ball is stable underφ, so by Brouwer’s fixed point theorem,φhas

a fixed point therein.