astronomical Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxi ...

and

geodetic Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...

calculations, Cracovians are a clerical convenience introduced in the 1930s by Tadeusz Banachiewicz for solving systems of linear equations by hand. Such systems can be written as in 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

transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

of ''A'', ''A''^T, and multiplying the columns of ''A''^T by the column x. This amounts to the definition of a new type of

matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

denoted here by '∧'. Thus . The Cracovian product of two matrices, say ''A'' and ''B'', is defined by , where ''B''^T and ''A'' are assumed compatible for the common ( Cayley) type of matrix multiplication. Since , the products and {{nowrap, 1=''A'' ∧ (''B'' ∧ ''C'') will generally be different; thus, Cracovian multiplication is non- associative. Cracovians are an example of a

quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...

. 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 computing, sequential access memory (SAM) is a class of data storage devices that read stored data in a sequence. This is in contrast to random access memory (RAM) where data can be accessed in any order. Sequential access devices are usually a ...

in computers of those times — mostly magnetic tape memory and

drum memory Drum memory was a magnetic data storage device invented by Gustav Tauschek in 1932 in Austria. Drums were widely used in the 1950s and into the 1960s as computer memory. For many early computers, drum memory formed the main working memory of ...

. 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.

In programming

In R the desired effect can be achieved via the crossprod() function. Specifically, the Cracovian product of matrices ''A'' and ''B'' can be obtained as crossprod(B, A).

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 is named after the author. * Kocinski, J. (2004). ''Cracovian Algebra'', Nova Science Publishers. Astrometry History of astronomy Matrix theory