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Factorizing matrices by Dirichlet multiplication. (English) Zbl 1263.15013
A Dirichlet multiplier is a matrix \[ \left [\begin{matrix} a_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_2 & a_1 & 0 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_3 & 0 & a_1 & 0 & 0 & 0 & 0 & 0 & \ldots \\ a_4 & a_2 & 0 & a_1 & 0 & 0 & 0 & 0 & \ldots \\ a_5 & 0 & 0 & 0 & a_1 & 0 & 0 & 0 & \ldots \\ a_6 & a_3 & a_2 & 0 & 0 & a_1 & 0 & 0 & \ldots \\ a_7 & 0 & 0 & 0 & 0 & 0 & a_1 & 0 & \ldots \\ a_8 & a_4 & 0 & a_2 & 0 & 0 & 0 & a_1 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \right] . \] where \(a_1, a_2, \ldots\) is a sequence of numbers.
Using finite submatrices of Dirichlet multipliers, the author shows that any nonsingular matrix is a product of Dirichlet multipliers. He also gives an efficient algorithm for a numerical factorization of such a matrix.

MSC:
15A23 Factorization of matrices
15B99 Special matrices
65F50 Computational methods for sparse matrices
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