astronomical Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include ...

and geodetic calculations, Cracovians are a clerical convenience introduced in 1925 by Tadeusz Banachiewicz for solving systems of

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

s by hand. Such systems can be written as in

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

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 (mathematics), 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 ...

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, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...

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 In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

. Cracovians are an example of a

quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...

. 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 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. Many early computers, called drum computers or drum machines, used drum ...

. Use of Cracovians in astronomy faded as computers with bigger

random access memory Random-access memory (RAM; ) is a form of electronic computer memory that can be read and changed in any order, typically used to store working data and machine code. A random-access memory device allows data items to be read or written ...

came into general use. Any modern reference to them is in connection with their non-associative multiplication. Named for recognition of the City of Cracow.

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 An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...

1751 is named after the author. * Kocinski, J. (2004). ''Cracovian Algebra'', Nova Science Publishers. Astrometry History of astronomy Matrix theory