Logical matrix: Difference between revisions

Revision as of 02:58, 14 September 2024

Revision as of 03:02, 14 September 2024

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Suppose <math>(P_i),\, i=1,2,\ldots,m</math> and <math>(Q_j),\, j=1,2,\ldots,n</math> are two logical vectors. The [[outer product]] of ''P'' and ''Q'' results in an ''m'' × ''n'' [[rectangular relation]]

:<math>m_{ij} = P_i \land Q_j.</math>

A reordering of the rows and columns of such a matrix can assemble all the ones into a rectangular part of the matrix.<ref name=GS>{{cite book | doi=10.1017/CBO9780511778810 | isbn={{Format ISBN|9780511778810}}978-0-511-77881-0 |first=Gunther |last=Schmidt | page=91 | title=Relational Mathematics | chapter=6: Relations and Vectors | publisher=Cambridge University Press | year=2013 | author-link=Gunther Schmidt }}</ref>

Let ''h'' be the vector of all ones. Then if ''v'' is an arbitrary logical vector, the relation ''R'' = ''v h''^T has constant rows determined by ''v''. In the [[calculus of relations]] such an ''R'' is called a vector.<ref name=GS/> A particular instance is the universal relation <math>hh^{\operatorname{T}}</math>.

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* {{cite encyclopedia |author-link=Richard A. Brualdi |first=Richard A. |last=Brualdi |title=Combinatorial Matrix Classes |publisher=Cambridge University Press |encyclopedia=Encyclopedia of Mathematics and its Applications |volume=108 |date=2006 |isbn=978-0-521-86565-4 |doi=10.1017/CBO9780511721182}}

* {{cite encyclopedia |first=Richard A. |last=Brualdi |first2=Herbert J. |last2=Ryser |title=Combinatorial Matrix Theory |publisher=Cambridge University Press |encyclopedia=Encyclopedia of Mathematics and its Applications |volume=39 |date=1991 |isbn=0-521-32265-0 |doi=10.1017/CBO9781107325708}}

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* {{Citation | last1=Kim | first1=Ki Hang|author-link=Ki-Hang Kim | title=Boolean Matrix Theory and Applications |year=1982| isbn=978-0-8247-1788-9}}

* {{cite journal |author-link=H. J. Ryser |first=H.J. |last=Ryser |title=Combinatorial properties of matrices of zeroes and ones |journal=[[Canadian Journal of Mathematics]] |volume=9 |issue= |pages=371–7 |date=1957 |doi= |url=}}