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The geometry of spectral interlacing. (English) Zbl 1500.15033

Let \(S\) be a \(n\times n\) Hermitian matrix and let \(\sigma_{o}(S)\) denote the ordered vector of \((\lambda_{1},\dots,\lambda_{n})\) of its eigenvalues with \(\lambda_{1}\leq \dots\leq \lambda_{n}\). A well-known theorem states (*): if \(\sigma_{o}(S)=(\lambda_{1},\dots,\lambda_{n})\) then for each vector \(v\in \mathbb{C}^{n}\) we have \(\sigma_{o}(S+vv^{\ast })=(\mu_{1},\dots,\mu_{n})\) where \(\lambda_{i}\leq \mu_{i}\) for each \(i\) and \(\mu_{i}\leq \lambda_{i+1}\) for \(i\neq n\); conversely, for any choice of \((\mu_{1},\dots,\mu_{n})\in \mathbb{R}\) satisfying these conditions there is an appropriate \(v\) (see, for example, [R. A. Horn and C. R. Johnson, Matrix analysis. 2nd ed. Cambridge: Cambridge University Press (2013; Zbl 1267.15001)]). The object of this paper is to explore such relationships of spectral interlacing further.
Fix a unitary matrix \(Q\) such that \(Q^{\ast }SQ=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})\). Then the \(i\)-th column \(Q\) is an eigenvector for the eigenvalue \(\lambda_{i}\) and we define \(\mathcal{O}_{Q}\) to be the set of all \(Qp\) where \(p\in \mathbb{R}^{n}\) has nonnegative entries. Define \(F: \mathcal{O}_{Q}\rightarrow \mathcal{P}_{F}\) by \(v\longmapsto \sigma_{o}(S+vv^{\ast })\) where \(\mathcal{P}_{F}:=[\lambda_{1},\lambda_{2}]\times \lbrack \lambda_{2},\lambda_{3}]\times \dots\times \lbrack \lambda_{n},\infty )\). Finally, for \(r>0\) let \(S(r):=\left\{ v\in \mathbb{C} ^{n}~|~\left\Vert v\right\Vert =r\right\} \) and \(\mathcal{P}_{F}^{r}:=\) \( \left\{ \mu \in \mathcal{P}_{F}~|~\sum_{j}\mu_{j}=r^{2}+\sum_{j}\lambda_{j}\right\}\). Then the restriction of \(F\) to vectors of length \(r\) defines a function \(F^{r}:\mathcal{O}_{Q}\cap S(r)\rightarrow \mathcal{P}_{F}^{r}\). The authors prove that \(F\) and the functions \(F^{r}\) are homeomorphisms and are diffeomorphisms between the interiors of the domain and image. Clearly, (*) is a consequence of this result. A similar result is proved for Cauchy’s theorem on the interlacing of the eigenvalues of \(S\) with those of the \((n+1)\times (n+1)\) matrices of the form \[ T(v,c):=\left[ \begin{array}{cc} S & v \\ v^{\ast } & c \end{array} \right] \text{.} \]

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A04 Linear transformations, semilinear transformations

Citations:

Zbl 1267.15001
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References:

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