On the dimension of the algebra generated by two positive semi-commuting matrices
Kandić, Marko ; Šivic, Klemen, matematik
Gerstenhaberjev izrek pravi, da je dimenzija enotske algebre, generirane z dvema komutirajočima ▫$n \times n$▫ matrikama, največ ▫$n$▫. Študiramo analog tega vprašanja za pozitivne matrike s ...pozitivnim komutatorjem. Dokažemo, da je dimenzija algebre, generirane s takima matrikama, največ ▫$\frac{n(n+1)}{2}$▫ in da je ta zgornja meja dosegljiva. Vprašanje obravnavamo tudi v primeru, ko je ena od matrik permutacijska ali pridružena, ali pa sta obe hkrati idempotentni. V teh primerih je zgornja meja bistveno manjša. Posebej, dimenzija algebre, generirane z dvema semi-komutirajočima pozitivnima matrikama, je največ 9. Ta zgornja meja je dosegljiva.
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