## Abstract

We show that the recent tensor-train (TT) decompositions of matrices come up from its recursive Kronecker-product representations with a systematic use of common bases. The names TTM and QTT used in this case stress the relation with multilevel matrices or quantization that increases artificially the number of levels. Then we investigate how the tensor-train ranks of a matrix can be related to those of its inverse. In the case of a banded Toeplitz matrix, we prove that the tensor-train ranks of its inverse are bounded above by 1+(l+u)_{2}, where l and u are the bandwidths in the lower and upper parts of the matrix without the main diagonal.

Original language | English |
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Pages (from-to) | 394-403 |

Number of pages | 10 |

Journal | Computational Methods in Applied Mathematics |

Volume | 11 |

Issue number | 3 |

Publication status | Published - 2011 |

Externally published | Yes |

## Keywords

- Banded matrices
- Inverse matrices
- Multilevel matrices
- QTT-ranks
- Tensor ranks
- Tensor-train decomposition
- Toeplitz matrices