{{for|people from the city of Cracow|Kraków}}
{{no footnotes|date=May 2015}} In [[astronomy|astronomical]] and [[geodesy|geodetic]] calculations, '''Cracovians''' are a clerical convenience introduced in 1925 by [[Tadeusz Banachiewicz]] for solving systems of [[linear equation]]s by hand. Such systems can be written as {{nowrap|1=''A'''''x''' = '''b'''}} in [[matrix (mathematics)|matrix]] notation where '''x''' and '''b''' are column vectors and the evaluation of '''b''' requires the multiplication of the rows of ''A'' by the vector '''x'''.
Cracovians introduced the idea of using the [[matrix transpose|transpose]] of ''A'', ''A''<sup>T</sup>, and multiplying the columns of ''A''<sup>T</sup> by the column '''x'''. This amounts to the definition of a new type of [[matrix multiplication]] denoted here by '∧'. Thus {{nowrap|1='''x''' ∧ ''A''<sup>T</sup> = ''b'' = ''A'''''x'''}}. The '''Cracovian product''' of two matrices, say ''A'' and ''B'', is defined by {{nowrap|1=''A'' ∧ ''B'' = ''B''<sup>T</sup>''A''}}, where ''B''<sup>T</sup> and ''A'' are assumed compatible for the common ([[Arthur Cayley|Cayley]]) type of matrix multiplication.
Since {{nowrap|1=(''AB'')<sup>T</sup> = ''B''<sup>T</sup>''A''<sup>T</sup>}}, the products {{nowrap|1=(''A'' ∧ ''B'') ∧ ''C''}} and {{nowrap|1=''A'' ∧ (''B'' ∧ ''C'')}} will generally be different; thus, Cracovian multiplication is non-[[associativity|associative]]. Cracovians are an example of a [[quasigroup]].
Cracovians adopted a column-row convention for designating individual elements as opposed to the standard row-column convention of matrix analysis. This made manual multiplication easier, as one needed to follow two parallel columns (instead of a vertical column and a horizontal row in the matrix notation.) It also sped up computer calculations, because both factors' elements were used in a similar order, which was more compatible with the [[sequential access memory]] in computers of those times — mostly [[magnetic tape data storage|magnetic tape memory]] and [[drum memory]]. Use of Cracovians in astronomy faded as computers with bigger [[random access memory]] came into general use. Any modern reference to them is in connection with their non-associative multiplication.
Named for recognition of the City of [[Kraków]].
==In programming== In [[R programming language|R]] the desired effect can be achieved via the <code>crossprod()</code> function. Specifically, the Cracovian product of matrices ''A'' and ''B'' can be obtained as <code>crossprod(B, A)</code>.
==References== *Banachiewicz, T. (1955). ''Vistas in Astronomy'', vol. 1, issue 1, pp 200–206. *Herget, Paul; (1948, reprinted 1962). ''The computation of orbits, University of Cincinnati Observatory'' (privately published). [[Asteroid]] [[1751 Herget|1751]] is named after the author. * Kocinski, J. (2004). ''Cracovian Algebra'', Nova Science Publishers.
[[Category:Astrometry]] [[Category:History of astronomy]] [[Category:Matrix theory]]