Cracovian

In astronomical and geodetic calculations, Cracovians are a clerical convenience introduced in 1925 by Tadeusz Banachiewicz for solving systems of linear equations by hand. Such systems can be written as Ax = b 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 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 denoted here by '∧'. Thus x ∧ A^T = b = Ax. The Cracovian product of two matrices, say A and B, is defined by A ∧ B = B^TA, where B^T and A are assumed compatible for the common (Cayley

) type of matrix multiplication.

Since (AB)^T = B^TA^T, the products (A ∧ B) ∧ C and A ∧ (B ∧ C) will generally be different; thus, Cracovian multiplication is non-

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

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 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 1751 is named after the author.
Kocinski, J. (2004). Cracovian Algebra, Nova Science Publishers.