Best rank k approximation for binary forms

• Autores: Giorgio Ottaviani, Alicia Tocino Sánchez
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 1, 2018, págs. 163-171
• Idioma: inglés
• DOI: 10.1007/s13348-017-0206-6
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• Resumen

  • In the tensor space {{\mathrm {Sym}}}^d \mathbb {R}^2 of binary forms we study the best rank k approximation problem. The critical points of the best rank 1 approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank k approximation problem lie in the same hyperplane. As a consequence, every binary form may be written as linear combination of its critical rank 1 tensors, which extends the Spectral Theorem from quadratic forms to binary forms of any degree. In the same vein, also the best rank k approximation may be written as a linear combination of the critical rank 1 tensors, which extends the Eckart–Young theorem from matrices to binary forms.

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