Integrators of matrices
type: Item
instance of: scholarly article
title: Integrators of matrices
zbMATH Open document ID: 1133.15007
author: B. V. Rajarama Bhat
author: Mithun Mukherjee
published in: Linear Algebra and its Applications
publication date: 2007-09-13T00:00:00Z
review text: It has been shown by \textit{R. Pereira} [J. Math. Anal. Appl. 285, 336--348 (2003; Zbl 1046.47002)] that all linear operators on finite dimensional Hilbert spaces admit trace vectors and hence they admit derivatives. In this paper the process is reversed. Given any operator \(B\) on an \((n-1)\)-dimensional Hilbert space \({\mathcal H}\), the authors try to find an operator \(A\) on an \(n\)-dimensional Hilbert space \({\mathcal H}'={\mathcal H} \oplus {\mathbb C}\), with orthogonal projection \(P\) having range \({\mathcal H}\), such that \(B=PAP|_{P({\mathcal H}')}\) and \(P\) is a differentiator of \(A\), i.e. \(p_B(x)=\frac{1}{n} p'_A(x)\). For the simplicity the space \({\mathcal H}\) is taken as \({\mathbb C}^{n-1}\), and \({\mathcal H}'\) is taken as \({\mathbb C}^n\). Then \(B\), \(A\) and \(P\) are matrices in \(M_{(n-1)} ({\mathbb C})\), \(M_n({\mathbb C})\), and \(M_n({\mathbb C})\), respectively, and they have the forms \[ A=\left[ \begin{matrix} B & u \\ v' & \tau(A) \end{matrix}\right], \quad P =\left[ \begin{matrix} I_{n-1} & 0 \\ 0 & 0 \end{matrix}\right] \] for some vectors \(u,v \in {\mathbb C}^{n-1}\), here \(\tau(A)=\frac 1{\dim {\mathcal H}} tr(A)\) is a normalized trace of \(A\). For a given matrix \(B\in M_{n-1}({\mathbb C})\) a matrix \(A\in M_n({\mathbb C})\) is said to be an \textit{integration} of \(B\) if it has the above block structure and \(p_B(x)=\frac{1}{n} p'_A(x)\). In this case the pair of vectors \((u,v)\) is said to be an \textit{integrator} of \(B\) and \(\det A\) is said to be the \textit{scalar} of integration. A square matrix \(B\) is said to be \textit{integrable} if it admits an integration. An integrable matrix \(B\) is said to be \textit{uniquely integrable} if there exists unique scalar \(\alpha\) such that for any \(A\) which is an integration of \(B\) it holds that \(\det(A) =\alpha\). An integrable matrix \(A\) is called \textit{freely integrable} if for any complex number \(\alpha\) there exists an integration \(A\…
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 26B15
zbMATH DE Number: 5190761
zbMATH Keywords: matrices
zbMATH Keywords: differentiators
zbMATH Keywords: integrators
zbMATH Keywords: non-derogatory
zbMATH Keywords: finite dimensional Hilbert spaces
zbMATH Keywords: trace vectors
zbMATH Keywords: derivatives
zbMATH Keywords: integration
zbMATH Keywords: integrable matrix
reviewed by: Alexander Guterman
MaRDI profile type: Publication
full work available at URL: j.laa.2007.04.003
OpenAlex ID: W2086013444
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DOI: 10.1016/J.LAA.2007.04.003
description: scientific article; zbMATH DE number 5190761