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Despre teorema lui Sylvester

 
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lasamasatelas
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Joined: 16 Nov 2007
Posts: 70

PostPosted: Sat May 21, 2011 7:05 pm    Post subject: Despre teorema lui Sylvester Reply with quote

\text{In caz ca nu stiti teorema lui Sylvester spune ca daca }A,B\text{ sunt 2 matrici de tip }m\times n\text{ si }n\times k\text{ atunci }\\ rang(AB)\geq rang(A)+rang(B)-n.

\text{Problema: Sa se arate ca asta e cel mai bun rezultat posibil. Mai precis, daca }a,b,c\text{ sunt 3 intregi a.i. }\\ 0\leq a\leq\min\[ m,n\},\, 0\leq b\leq\min\[ n,k\}\text{ si }\max\{ 0,a+b-n\}\leq c\leq\min\[ a,b\}\\ \text{atunci exista 2 matrici }A,B\text{ de tip }m\times n\text{ si }n\times k\text{ a.i. }rang(A)=a,\, rang(B)=b,\, rang(AB)=c.

\text{Apropo de Sylvester, acesta are o generalizare, a lui Frobenius. Daca }A,B,C\text{ sunt 3 matrici atunci}\\ rang(AB)+rang(BC)\leq rang(B)+rang(ABC)\\ \text{Daca luam }B=I_n\text{ (matricea identitate) se obtine Sylvester pentru }A,C.

EDIT Tocmai am facut o mica corectie. Acolo unde acum e min{a,b} pusesem initial min{m,k}. (Era gresit.)
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