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The geometry of spectral interlacing

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Abstract

We provide a detailed description of the maps associated with spectral interlacing in two scenarios, for rank one perturbations and bordering of symmetric and Hermitian matrices. The arguments rely on standard techniques of nonlinear analysis.

Introduction

We recall two standard results, presented in more detail in [7]. Endow Cn with the Euclidean inner product , which is anti-linear in the first coordinate. For vCn, the linear rank one map vv=vv is uv,uv. Let S be an n×n Hermitian matrix, with ordered eigenvalues λ1λn.

Theorem 1.1 Cor. 4.3.9, Th. 4.3.21 [7]

For vCn, let the eigenvalues of T=T(v)=S+vv be μ1μn. Then the eigenvalues of S and T interlace,λ1μ1λ2μ2λnμn. Conversely, for a sequence {μj} interlacing {λk} as above, there is vCn for which the eigenvalues of T=S+vv are {μj}.

The second result is Cauchy's interlacing theorem.

Theorem 1.2 Th. 4.3.17 [7]

Let vCn, cR, μ1μ2μn+1 be the eigenvalues of the bordered matrixT=T(v,c)=(Svvc). Then the eigenvalues of S and T interlace,μ1λ1μ2λ2μ3λnμn+1. For a sequence {μj} interlacing {λk}, there exist vCn and c>0 for which the eigenvalues of T=T(v,c) are {μj}.

The results can be found in essentially any advanced book on linear algebra. Demmel [2] quotes Löwner's formula [15] for the vector v in terms of the spectra of S and T in Theorem 1.1. Extensions abound. Ionascu [9] indicates a number of references (from which we emphasize [20]) proving related results for compact self-adjoint operators S on a separable Hilbert space. Simon [18] considers finer aspects of the spectrum of rank one perturbations of (mostly) Schrödinger operators with very interesting applications, which are beyond the scope of this text.

In this text, we cast these results in geometric terms. Fix a normalized eigenbasis Q=[q1,,qn] of S, arranged as columns of a unitary matrix Q. Let OQCn, the positive orthant associated with Q, be the set of vectors of the form Qp, where pRn has nonnegative entries.

A vector vRn is ordered if v1vn. Write σo(S)Rn for the ordered vector with entries given by the eigenvalues of S. For r>0, define the polytopesPF=[λ1,λ2]×[λ2,λ3]××[λn,),PG=(,λ1]×[λ1,λ2]×[λ2,λ3]××[λn,), two half-open boxes, and the sphere S(r)={vCn,v=r}.

Theorem 1.3

Let S be an n×n Hermitian matrix, with spectrum λ1<<λn, an eigenbasis Q and positive orthant OQ. The following maps are homeomorphisms, and diffeomorphisms between the interior of their domain and image.F:DF=OQPF,G:DG=OQ×RPGvσo(S+vv)(v,c)σo(T(v,c)) and their restrictionsFr:DFr=OQS(r)PFr=PF{μRn,jμj=r2+jλj},Gc:OQ×{c}PGc=PG{μRn+1,jμj=kλk+c},Gr,c:(OQ(S(r))×{c}PGc{μRn+1,jμj2=kλk2+2r2+c2}.

If S has distinct eigenvalues, Theorem 1.1, Theorem 1.2 then follow. Full generality is attained by taking limits.

The general rank one Hermitian perturbation matrix is of the form cvv for a real unit vector v and cR. The sign of c specifies if the perturbation of S pushes the spectrum to the right (the case c>0) or to the left (c<0). For the results above, c0: minor alterations handle c0. Clearly, the interlacing property is associated with the geometry of the polytopes PF and PG.

It is rather intriguing that the interior of an orthant OQ is taken by F to PF, a closed box with a face removed. As we shall see in the proof, some faces of OQ are creased by F, giving rise to two faces of PQ. Something similar happens with G, but now PG is a box with two faces removed.

This is what happens for n=2. The horizontal axis is taken to the union of a horizontal and a vertical segment. The vertical axis is sent to itself.

The simple geometry of the maps F and G has implications to the computation of their inverses, frequently described as an inverse problem ([7]). For F, given a symmetric matrix S with ordered, simple, spectrum λ and an interlacing ordered n-tuple μ, we look for a rank one perturbation cvv such that the spectrum of S+cvv is μ. Theorem 1.3 shows that, in principle, the problem is solvable by numerical continuation starting from any interior point of DF, as there are no critical values there. The same argument proves that continuation from an interior point of DG obtains the inverse of G.

Given a function f:XY, the preimages of yY are the points in the set f1(y)={xX,f(x)=y}. We now consider the preimages of the maps in the previous theorem. We have to distinguish between matrices with real or complex entries. Let K=R or C and defineabs:KnOQ,v=j=1ncjqjj=1n|cj|qj.

Theorem 1.4

Let S as in the previous theorem.

  • 1.

    Say the entries of S lie in K=R or C, vKn. Then F and G extend toFˆ:KnPF,Gˆ:Kn×RPG.vσo(S+vv)(v,c)σo(T(v,c)) Moreover, Fˆ(v)=Fˆ(w)Gˆ(v)=Gˆ(w)abs(v)=abs(w). In particular, all preimages of a point belong to the same sphere S(r).

  • 2.

    If K=R, a point μPF belonging to exactly k faces has 2nk preimages under Fˆ or Gˆ. If K=C, for both functions the preimages of μ form a product of nk1 circles: a torus.

Recently, Maciazek and Smilansky [16] considered analogous inverse problems and pointed out the relevance of discrete information provided by strings of signs. We believe our presentation sheds some light on the issue.

Theorem 1.3, Theorem 1.4 are a strengthened version of a very special case of the celebrated Horn's conjecture [6], whose resolution, after work by several authors ([10], [8], [12], [13]), is beautifully described in [11]. The conjecture answers a question by Weyl [21]: what are the possible spectra of the sum A+B of two Hermitian matrices of given spectrum? Horn originally provided a list of linear inequalities on the eigenvalues of the three matrices which provide necessary and sufficient conditions relating their spectra. For A=S, B=vv with v=r, Horn's conjecture states that the image of the map Fr is indeed PFr.

Part of the statements in Theorem 1.3, Theorem 1.4 may be deduced from the sophisticated machinery of symplectic geometry. To give an idea of a more familiar context, the Schur-Horn theorem for Hermitian matrices [5] is a consequence of the powerful theorems about the convexity of the image of moment maps of torus actions by Atiyah ([1]) and Guillemin-Sternberg ([4]). The result for symmetric matrices then follows by an argument by Duistermaat ([3]). Similarly, the surjectivity of the maps F,Fr and G also follow from convexity arguments, once the appropriate symplectic setting is identified. Here, we take what Thompson ([19]) calls a low road in linear algebra. The fact that the critical set of the maps in Theorem 1.3 is restricted to the boundary of their domains (and in particular continuation to the interior is safe) does not follow directly from rote application of these more general results.

The proof of Theorem 1.3 relies on a combination of well known facts of real analysis, condensed in Lemma 2.1. The verification of the hypotheses of the lemma is somewhat different for F and G. In both cases, the theorem is proved by induction on the dimension. In the inductive step, we see how faces of the domain are ‘creased’ by either F or G so as to obtain the faces of the image parallelotope. Theorem 1.4 is a simple consequence of Theorem 1.3.

The authors are grateful to an anonymous reader of a previous version of this text, who indicated errors and suggested a number of improvements.

Section snippets

A real analysis lemma

The outline of the proof of Theorem 1.3 is the same for the functions F:DFPF and G:DGPG. In a nutshell, we must check the hypotheses of the lemma below, which combines familiar arguments from real analysis. We state it so as it applies directly to F. Let D be OI, the closed positive orthant of Rn, and P be PF. Denote by int X the interior of a set X.

Lemma 2.1

Let H˜:DRn be a function satisfying the following properties.

  • (H1)

    H˜ is a continuous, proper map, i.e., limvH˜(v)=.

  • (H2)

    The restriction of H˜ to

Proof of Theorem 1.3 for F and Fr

Without loss, suppose S=D, a diagonal matrix with eigenvaluesD11=λ1<<Dnn=λn. We then take Q=[e1,,en] to consist of the canonical vectors, so that Q=I and OIRn is the usual positive orthant. ConsiderF˜:D=DF=OIRn,vσo(D+vv) where now all numbers in sight are real. Complex numbers will return only in the proof of Theorem 1.4 in Section 5.

The set D consists of n faces of D=OI,Ei={vRn,vi=0},i=1,,n.

The parallelotope PRn has 2n1 faces, which we now describe. SetLi=[λ1,λ2]××[λi1,λi],Ri=[λi,λ

Proof of Theorem 1.3 for G, Gc and Gr,c

Again, without loss, S=D, a diagonal matrix with eigenvalues λ1<<λn. Now D=OI×R has faces of the form Ei×R and the boxP=PG=(,λ1]×[λ1,λ2]×[λ2,λ3]××[λn,)Rn+1 has 2n faces. Recall that all numbers in sight are real. In particular, v is the transpose of v. DefineT=T(v,c)=(Dvvc).

We must show that the map G˜:DRn+1,(v,c)σo(T(v,c))=σo(T) defines a homeomorphism G:DP. This time, as we shall see, G takes every face of D to two adjoining faces of P.

As before Dd consists of the points (v,c)D

Proof of Theorem 1.4

We consider Fˆ:KnPF, the other cases being analogous.

From Theorem 1.3, as F is injective, F(abs(v))=F(abs(w)) if and only if v=w. From the surjectivity of F, given μPF, there is a (unique) vOQ for which F(v)=μ. Hence, Fˆ1(μ)=abs1F1(μ)=abs1(v).

Each nonzero coordinate vk of v gives rise to a circle eiθkvk of possible values for the k-th coordinate of abs1(v). Clearly zk=0 if and only if zEk.  

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The authors are supported by CNPq grant number: 306.309/2016-5, CAPES and FAPERJ grant number: E-26/202.908;2018.

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