Two applications of the theory of primary matrix functions
A systematic exposition of primary matrix functions was given by \textit{R. A. Horn} and \textit{C. R. Johnson} [Topics in matrix analysis, Cambridge University Press, New York (1991; Zbl 0729.15001)]. For a complex \(n\times n\) matrix the authors use the Jordan canonical form \[ A=S\left[ \begin{matrix} J_{n_{1}}(\lambda) & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & J_{n_{d}}(\lambda) \end{matrix} \right] S^{-1} \] where \(J_{k}(\lambda) \) are \(k\)-Jordan blocks. Assume that \(f:D\rightarrow\mathbb{C}\), \(D\subset\mathbb{C}\), is a given analytic function in an open set containing the spectrum of \(A.\) The primary matrix function is defined by \[ f\left( A\right) =S\left[ \begin{matrix} f(J_{n_{1}}(\lambda)) & 0 & 0\\ 0 & \ddots & 0\\ 0 & 0 & f(J_{n_{d}}(\lambda)) \end{matrix} \right] S^{-1} \] in which each block \(f( J_{k}( \lambda)) \) is an upper triangular Toeplitz matrix with the entries \(f( \lambda) ,\) \(f'( \lambda) ,\dots,f^{( k-1)}( \lambda) /( k-1)!\) in the first row. The authors consider whether \(f(A) \) can have rank one, and whether, for a given \(B,\) there is a unique \(A\) such that \(f(A) =B.\)
scientific article; zbMATH DE number 1880049