Sylvester domain
In mathematics, a Sylvester domain, named after James Joseph Sylvester by {{harvtxt|Dicks|Sontag|1978}}, is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix, and B is an n by s matrix over R, then
:ρ(AB) ≥ ρ(A) + ρ(B) – n
where ρ is the inner rank of a matrix. The inner rank of an m by n matrix is the smallest integer r such that the matrix is a product of an m by r matrix and an r by n matrix.
{{harvtxt|Sylvester|1884}} showed that fields satisfy Sylvester's law of nullity and are, therefore, Sylvester domains.
References
- {{Citation | last2=Sontag | first2=Eduardo D. | last1=Dicks | first1=Warren | title=Sylvester domains | doi=10.1016/0022-4049(78)90011-7 |mr=509164 | year=1978 | journal=Journal of Pure and Applied Algebra | issn=0022-4049 | volume=13 | issue=3 | pages=243–275| doi-access=free }}
- {{Citation | last1=Sylvester | first1=James Joseph | title=On involutants and other allied species of invariants to matrix systems | url=https://books.google.com/books?id=7zw9AAAAIAAJ&pg=PA133 | id=Reprinted in collected papers volume IV, paper 15 | year=1884 | journal=Johns Hopkins University Circulars | volume=III | pages=9–12, 34–35}}
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