Regular ArticlesThe geometry of spectral interlacing☆
Introduction
We recall two standard results, presented in more detail in [7]. Endow with the Euclidean inner product which is anti-linear in the first coordinate. For , the linear rank one map is . Let S be an Hermitian matrix, with ordered eigenvalues .
Theorem 1.1 Cor. 4.3.9, Th. 4.3.21 [7] For , let the eigenvalues of be . Then the eigenvalues of S and T interlace, Conversely, for a sequence interlacing as above, there is for which the eigenvalues of are .
The second result is Cauchy's interlacing theorem.
Theorem 1.2 Th. 4.3.17 [7] Let , , be the eigenvalues of the bordered matrix Then the eigenvalues of S and T interlace, For a sequence interlacing , there exist and for which the eigenvalues of are .
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 of S, arranged as columns of a unitary matrix Q. Let , the positive orthant associated with , be the set of vectors of the form Qp, where has nonnegative entries.
A vector is ordered if . Write for the ordered vector with entries given by the eigenvalues of S. For , define the polytopes two half-open boxes, and the sphere . Theorem 1.3 Let S be an Hermitian matrix, with spectrum , an eigenbasis and positive orthant . The following maps are homeomorphisms, and diffeomorphisms between the interior of their domain and image. and their restrictions
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 for a real unit vector v and . The sign of c specifies if the perturbation of S pushes the spectrum to the right (the case ) or to the left (). For the results above, : minor alterations handle . Clearly, the interlacing property is associated with the geometry of the polytopes and .
It is rather intriguing that the interior of an orthant is taken by F to , a closed box with a face removed. As we shall see in the proof, some faces of are creased by F, giving rise to two faces of . Something similar happens with G, but now is a box with two faces removed.
This is what happens for . 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 such that the spectrum of is μ. Theorem 1.3 shows that, in principle, the problem is solvable by numerical continuation starting from any interior point of , as there are no critical values there. The same argument proves that continuation from an interior point of obtains the inverse of G.
Given a function , the preimages of are the points in the set . We now consider the preimages of the maps in the previous theorem. We have to distinguish between matrices with real or complex entries. Let or and define
Theorem 1.4 Let S as in the previous theorem. Say the entries of S lie in or , . Then F and G extend to Moreover, . In particular, all preimages of a point belong to the same sphere . If , a point belonging to exactly k faces has preimages under or . If , for both functions the preimages of μ form a product of 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 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 , with , Horn's conjecture states that the image of the map is indeed .
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 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 and . 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 be , the closed positive orthant of , and be . Denote by int X the interior of a set X.
Lemma 2.1 Let be a function satisfying the following properties. is a continuous, proper map, i.e., . The restriction of to
Proof of Theorem 1.3 for F and
Without loss, suppose , a diagonal matrix with eigenvalues We then take to consist of the canonical vectors, so that and is the usual positive orthant. Consider 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 consists of n faces of ,
The parallelotope has faces, which we now describe. Set
Proof of Theorem 1.3 for G, and
Again, without loss, , a diagonal matrix with eigenvalues . Now has faces of the form and the box has 2n faces. Recall that all numbers in sight are real. In particular, is the transpose of v. Define
We must show that the map defines a homeomorphism . This time, as we shall see, G takes every face of to two adjoining faces of .
As before consists of the points
Proof of Theorem 1.4
We consider , the other cases being analogous.
From Theorem 1.3, as F is injective, if and only if . From the surjectivity of F, given , there is a (unique) for which . Hence, .
Each nonzero coordinate of v gives rise to a circle of possible values for the k-th coordinate of . Clearly if and only if . □
References (21)
High, low, and quantitative roads in linear algebra
Linear Algebra Appl.
(1992)Convexity and commuting Hamiltonians
Bull. Lond. Math. Soc.
(1982)Applied Numerical Linear Algebra
(1997)The momentum map
- et al.
Convexity properties of moment mappings
Invent. Math.
(1981) Doubly stochastics matrices and the diagonal of a rotation matrix
Am. J. Math.
(1954)Eigenvalues of sums of Hermitian matrices
Pac. J. Math.
(1962)- et al.
Matrix Analysis
(2013) - et al.
Eigenvalue inequalities and Schubert calculus
Math. Nachr.
(1995) Rank-one perturbations of diagonal operators
Integral Equ. Oper. Theory
(2001)
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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.