Logical matrix: Difference between revisions

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Adding up all the ones in a logical matrix may be accomplished in two ways: first summing the rows or first summing the columns. When the row sums are added, the sum is the same as when the column sums are added. In [[incidence geometry]], the matrix is interpreted as an [[incidence matrix]] with the rows corresponding to "points" and the columns as "blocks" (generalizing lines made of points). A row sum is called its ''point degree'', and a column sum is the ''block degree''. The sum of point degrees equals the sum of block degrees.<ref name=BJL>E.g., see {{cite book |first1=Thomas |last1=Beth |first2=Dieter |last2=Jungnickel |author-link2=Dieter Jungnickel |first3=Hanfried |last3=Lenz |author-link3=Hanfried Lenz |title=Design Theory |publisher=[[Cambridge University Press]] |page=18 |year=1999 |edition=2nd |ISBN=978-0-521-44432-3}}</ref>

An early problem in the area was "to find necessary and sufficient conditions for the existence of an [[incidence structure]] with given point degrees and block degrees; (or in matrix language, for the existence of a (0,&thinsp;1)-matrix of type ''v''&nbsp;×&nbsp;''b'' with given row and column sums".<ref name=BJL/> This problem is solved by the [[Gale–Ryser theorem]].

==See also==