Thick compressions of thin algebras

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

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.

• Jezik: Angleški jezik
• Ključne besede: matrix algebra, compression, nonderogatory matrix, operator system, quantum clique, diagonal algebra, idempotent
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.01.2027
• Leto izida: 2027
• Št. strani: Str. 273-298
• Številčenje: Vol. 753
• PID: 20.500.12556/RUL-188788
• UDK: 512:517.9
• ISSN pri članku: 0024-3795
• DOI: 10.1016/j.laa.2026.09.017
• COBISS.SI-ID: 292670723
• Datum objave v RUL: 28.09.2026

Gradivo je del revije

• Naslov: Linear algebra and its applications
• Skrajšan naslov: Linear algebra appl.
• Založnik: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: matrična algebra, kompresija, nederogatorna matrika, operatorski sistem, kvantna klika, diagonalna algebra, idempotent