Circulants, displacements and decompositions of matrices
Following \textit{T. Kailath}, \textit{S.-Y. Kung} and \textit{M. Morf} [J. Math. Anal. Appl. 68 395-407 (1979; Zbl 0433.15001)] the authors introduce in \(\mathbb{C}^{n\times n}\) the linear operator, which transforms each matrix \(A\) in its displacement \(\nabla_ 0(A)=A-Z_ 0AZ^ T_ 0\), where \[ Z_ 0=\left[\begin{matrix} 0 &\ldots &\ldots &\ldots &0 \\ 1 &0 & & &\vdots \\ 0 & 1 &\ddots & &\vdots \\ \vdots &\ddots &\ddots & \ddots & \vdots \\ 0 &\ldots & 0 & 1 & 0 \end{matrix} \right] \] is the lower shift matrix. New formulas for the representation of matrices and their inverses in the form of sums of products of factor circulants are suggested, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalize the formulas of \textit{I. C. Gohberg} and \textit{A. A. Semencul} [Mat. Issled. 7, No. 2(24), 201-223 (1972; Zbl 0288.15004)], \textit{A. Ben-Artzi} and \textit{T. Shalom} [Linear Algebra Appl. 75, 173-192 (1986; Zbl 0586.15005)], and \textit{G. Heinig} and \textit{K. Rost} [Algebraic methods for Toeplitz-like matrices and operators (Akademie-Verlag, Berlin, 1984; Zbl 0549.15013)] and are useful for complexity analysis.
scientific article; zbMATH DE number 222146