Integrators of matrices
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 \(A\) of \(B\) such that \(\det(A)=\alpha\). The authors investigate these classes of matrices. In particular, it is shown that non-derogatory matrices are freely integrable. In addition the considerable number of examples of non-integrable matrices is presented.
scientific article; zbMATH DE number 5190761