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Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition
Abstract
The canonical decomposition of higher-order tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.-J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136--1155]. A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.
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Cited by
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
Oseledets, I. V. (2010) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/090757861 · Cited by 174
Metadata
- Authors
- De Lathauwer, L., De Moor, B., & Vandewalle, J.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2004-01
- Volume
- 26
- Issue
- 2
- Pages
- 295-327
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W2078789945
- Cited by
- 229
- References
- 0
- FWCI
- 5.05
- Type
- journal article
- Language
- en
- Open access
- closed
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- De Lathauwer, L., De Moor, B., & Vandewalle, J. (2004). Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition. SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/s089547980139786x
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