# Cracovian

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For people from the city of Cracow, see [Kraków](/source/Krak%C3%B3w).

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In [astronomical](/source/Astronomy) and [geodetic](/source/Geodesy) calculations, **Cracovians** are a clerical convenience introduced in 1925 by [Tadeusz Banachiewicz](/source/Tadeusz_Banachiewicz) for solving systems of [linear equations](/source/Linear_equation) by hand. Such systems can be written as *A***x** = **b** in [matrix](/source/Matrix_(mathematics)) 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](/source/Matrix_transpose) 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](/source/Matrix_multiplication) denoted here by '∧'. Thus **x** ∧ *A*T = *b* = *A***x**. The **Cracovian product** of two matrices, say *A* and *B*, is defined by *A* ∧ *B* = *B*T*A*, where *B*T and *A* are assumed compatible for the common ([Cayley](/source/Arthur_Cayley)) type of matrix multiplication.

Since (*AB*)T = *B*T*A*T, the products (*A* ∧ *B*) ∧ *C* and *A* ∧ (*B* ∧ *C*) will generally be different; thus, Cracovian multiplication is non-[associative](/source/Associativity). Cracovians are an example of a [quasigroup](/source/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](/source/Sequential_access_memory) in computers of those times — mostly [magnetic tape memory](/source/Magnetic_tape_data_storage) and [drum memory](/source/Drum_memory). Use of Cracovians in astronomy faded as computers with bigger [random access memory](/source/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](/source/Krak%C3%B3w).

## In programming

In [R](/source/R_programming_language) 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](/source/Asteroid) [1751](/source/1751_Herget) is named after the author.

- Kocinski, J. (2004). *Cracovian Algebra*, Nova Science Publishers.

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