Logical matrix: Difference between revisions

Revision as of 07:44, 16 January 2023

Revision as of 07:47, 16 January 2023

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==Row and column sums==

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''. PropositionThe 1.6sum inof ''Designpoint degrees equals the sum of block Theory''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> says that the sum of point degrees equals the sum of block degrees.

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]].