Factorization of CP-rank- 3 completely positive matrices

Jan Brandts; Michal Křížek

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 955-970
  • ISSN: 0011-4642

Abstract

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A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = B B ⊤ . If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A . In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3 . Failure of this algorithm implies that A does not have cp-rank 3 . Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.

How to cite

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Brandts, Jan, and Křížek, Michal. "Factorization of CP-rank-$3$ completely positive matrices." Czechoslovak Mathematical Journal 66.3 (2016): 955-970. <http://eudml.org/doc/286780>.

@article{Brandts2016,
abstract = {A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A=BB^\top $. If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$. In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$. Failure of this algorithm implies that $A$ does not have cp-rank $3$. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.},
author = {Brandts, Jan, Křížek, Michal},
journal = {Czechoslovak Mathematical Journal},
keywords = {completely positive matrix; cp-rank; factorization; discrete maximum principle},
language = {eng},
number = {3},
pages = {955-970},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Factorization of CP-rank-$3$ completely positive matrices},
url = {http://eudml.org/doc/286780},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Brandts, Jan
AU - Křížek, Michal
TI - Factorization of CP-rank-$3$ completely positive matrices
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 955
EP - 970
AB - A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A=BB^\top $. If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$. In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$. Failure of this algorithm implies that $A$ does not have cp-rank $3$. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.
LA - eng
KW - completely positive matrix; cp-rank; factorization; discrete maximum principle
UR - http://eudml.org/doc/286780
ER -

References

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  2. Berman, A., Shaked-Monderer, N., Completely Positive Matrices, World Scientific, River Edge (2003). (2003) Zbl1030.15022MR1986666
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  4. Berg, M. de, Kreveld, M. van, Overmars, M., Schwarzkopf, O., 10.1007/978-3-662-04245-8_1, Springer, Berlin (2000). (2000) MR1763734DOI10.1007/978-3-662-04245-8_1
  5. Post, K. A., 10.1007/BF01667408, Geom. Dedicata (1993), 45 115-120. (1993) Zbl0770.51021MR1199733DOI10.1007/BF01667408
  6. Shaked-Monderer, N., A note on upper bounds on the cp-rank, Linear Algebra Appl. 431 2407-2413 (2009). (2009) Zbl1180.15028MR2563031
  7. Steinhaus, H., One Hundred Problems in Elementary Mathematics, Popular Lectures in Mathematics 7 Pergamon Press, Oxford (1963). (1963) Zbl0116.24102MR0157881
  8. Sullivan, J. M., Polygon in a triangle: Generalizing theorem by Post, Preprint available at http://torus.math.uiuc.edu/jms/Papers/post.pdf (1996). (1996) 
  9. Vejchodský, T., Šolín, P., 10.1090/S0025-5718-07-02022-4, Math. Comput. 76 1833-1846 (2007). (2007) Zbl1125.65108MR2336270DOI10.1090/S0025-5718-07-02022-4

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