Cracovian

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For people from the city of Crakow, see Kraków.

Cracovians were defined and introduced into astronomic and geodesic calculations by Tadeusz Banachiewicz in the 1930s as a clerical convenience in solving systems of linear equations by hand. Such systems can be written as CX=x in matrix notation where X and x are column vectors and the evaluation of x requires the multiplication of a row of C by the column X.

Cracovians introduced the idea of using the transpose of C, C',and multiplying the columns of C' by the column X. This amounts to the definition of a new (yet another) type of matrix multiplication denoted here by '∧'. Thus CX=x=X∧C'. In general with arrays A and B, A∧B=B'A, B' and A being assumed compatible for the common (Cayley) type of matrix multiplication.

Since (AB)'=B'A' ⇒ (A∧B)∧C≠A∧(B∧C) Cracovian multiplication is non-associative. Actually any type of matrix multiplication which involves a transpose will be non-associative.

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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 speeded up computer calculations, because both factors' elements were used in the similar order, which was more compatible with the sequential access memory in computers of those times — mostly 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.

[edit] References

  • T.Banachiewicz (1955). Vistas in Astronomy, vol. 1, issue 1, pp 200-206.
  • Paul Herget (1948, reprinted 1962). The computation of orbits, University of Cincinnati Observatory (privately published). Asteroid 1751 is named after the author.
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