Abstract
We introduce the notion of \(J\)-Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by \(J\)-Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and \(\tau \) matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by \(J\)-Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by \(J\)-Hermitianity. Even if such hypothesis on the structure is not necessary in the case of algebras generated by one matrix, as it has been proved in [3], our result is relevant because it could yield new low complexity displacement formulas involving not one-matrix-generated commutative algebras.
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Notes
It is enough to consider a real optimal skeleton decomposition of the matrix \(LJAJM = \sum {{{{\tilde {x}}}_{m}}\tilde {y}_{m}^{t}} \) and then consider the skeleton decomposition of the matrix \(A = \sum {J{{L}^{{ - 1}}}{{{\tilde {x}}}_{m}}\tilde {y}_{m}^{t}{{M}^{{ - 1}}}J} \).
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Funding
This work has been partially supported by INdAM-GNCS, Project ASDRID supported by Rome Tor Vergata University, MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, Grant/Award number: CUP E83C18000100006.
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Bozzo, E., Deidda, P. & Di Fiore, C. Algebras Closed by J-Hermitianity in Displacement Formulas. Comput. Math. and Math. Phys. 61, 674–683 (2021). https://doi.org/10.1134/S0965542521050055
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DOI: https://doi.org/10.1134/S0965542521050055