Cracovian
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- For people from the city of Crakow, see Kraków.
Cracovians were introduced into astronomy 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 (mathematics) 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.
Cracovians adopted a column-row convention for designating individual elements as opposed to the standard row-column convention of matrix analysis. Use of Cracovians in astronomy faded as computers came into general use. Any modern reference to them is in connection with their non-associative multiplication.
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.
