Thick compressions of thin algebras

Omladič, Matjaž (author), Radjavi, Heydar (author), Šivic, Klemen (author), Zalar, Aljaž (author)

Let ${\mathcal A}$ be the algebra generated by a nonderogatory matrix in $M_{n^2}({\mathbb F})$, where ${\mathbb F} = {\mathbb C}$ or ${\mathbb R}$. We prove that there exist matrices $V \in M_{n^2,n}({\mathbb F})$ and $W \in M_{n,n^2}({\mathbb F})$ such that $WV = I_n$ and $W{\mathcal A}V = M_n({\mathbb F})$. Over ${\mathbb C}$, this is equivalent to the existence of a rank $n$ orthogonal projection $P$ for which $P{\mathcal A}P|_{{Rm Im} P} = M_n({\mathbb C}$. As a consequence, the diagonal algebra in $M_{n^2}({\mathbb F})$ has a quantum $n$-clique. The corresponding optimal result for the complex diagonal algebra was recently obtained independently by Weaver. Over ${\mathbb R}$, ordinary orthogonal compressions are generally too restrictive, and the corresponding geometric result is formulated using general rank-$n$ idempotents, possibly oblique. We also give an elementary two-sided construction for the diagonal algebra over both fields and determine the dimensions of the other three block spaces associated with a maximal orthogonal compression of a commutative self-adjoint algebra.

• Language: English
• Keywords: matrix algebra, compression, nonderogatory matrix, operator system, quantum clique, diagonal algebra, idempotent
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2027
• Year: 2027
• Number of pages: Str. 273-298
• Numbering: Vol. 753
• PID: 20.500.12556/RUL-188788
• UDC: 512:517.9
• ISSN on article: 0024-3795
• DOI: 10.1016/j.laa.2026.09.017
• COBISS.SI-ID: 292670723
• Publication date in RUL: 28.09.2026

Record is a part of a journal

• Title: Linear algebra and its applications
• Shortened title: Linear algebra appl.
• Publisher: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: matrična algebra, kompresija, nederogatorna matrika, operatorski sistem, kvantna klika, diagonalna algebra, idempotent