TY - JOUR

T1 - Congruence of matrix spaces, matrix tuples, and multilinear maps

AU - Belitskii, Genrich R.

AU - Futorny, Vyacheslav

AU - Muzychuk, Mikhail

AU - Sergeichuk, Vladimir V.

N1 - Funding Information:
V. Futorny was supported by the CNPq ( 304467/2017-0 ) and the FAPESP ( 2018/23690-6 ). V.V. Sergeichuk was supported by FAPESP ( 2018/24089-4 ). The work was started when V.V. Sergeichuk visited the Ben-Gurion University of the Negev.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - Two matrix vector spaces V,W⊂Cn×n are said to be equivalent if SVR=W for some nonsingular S and R. These spaces are congruent if R=ST. We prove that if all matrices in V and W are symmetric, or all matrices in V and W are skew-symmetric, then V and W are congruent if and only if they are equivalent. Let F:U×…×U→V and G:U′×…×U′→V′ be symmetric or skew-symmetric k-linear maps over C. If there exists a set of linear bijections φ1,…,φk:U→U′ and ψ:V→V′ that transforms F to G, then there exists such a set with φ1=…=φk.

AB - Two matrix vector spaces V,W⊂Cn×n are said to be equivalent if SVR=W for some nonsingular S and R. These spaces are congruent if R=ST. We prove that if all matrices in V and W are symmetric, or all matrices in V and W are skew-symmetric, then V and W are congruent if and only if they are equivalent. Let F:U×…×U→V and G:U′×…×U′→V′ be symmetric or skew-symmetric k-linear maps over C. If there exists a set of linear bijections φ1,…,φk:U→U′ and ψ:V→V′ that transforms F to G, then there exists such a set with φ1=…=φk.

KW - Congruence

KW - Multilinear maps

KW - Weak congruence

UR - http://www.scopus.com/inward/record.url?scp=85091332759&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.09.018

DO - 10.1016/j.laa.2020.09.018

M3 - Article

AN - SCOPUS:85091332759

SN - 0024-3795

VL - 609

SP - 317

EP - 331

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -