Cracovian Algebra

A rectangular cracovian is an n × n array {a_ij}, where (in contrast to matrix notation) a_ij denotes an entry in column iand row j. The braces are used to distinguish such a table of elements from a matrix of the same elements. The product of two cracovians A and B, defined only when two cracovians have the same number of rows, involves dot products of columns of A and B: The element in column k and row l of the product A∙B is given as

where the summation over the index i extends over all rows. Thus the product of cracovians is noncommutative and nonassociative.

Cracovian algebra was introduced in 1923-4 and developed over the next thirty years by the astronomer Thaddeus Banachiewicz at the Jagellonian University in Cracow, from which the name of this theory derives. The main applications of this algebra have been to theoretical astronomy, geodesy, and to least squares problems. The existence of homomorphisms between certain quasigroups and square cracovians having inverses suggests to Kociński the possible applicability of cracovian algebra to physics.

Most of the book describes a universe parallel to that usually described in linear algebra books: solutions of systems of linear equations (including a cracovian version of the Kronecker-Capelli theorem), the inverse cracovian, the decomposition of cracovians into elementary cracovians, the determinant of a square cracovian, rotation cracovians... The book ends with a ninety-three item bibliography and an index.

This is a monograph rather than a textbook. There are examples, but no exercises.

Those interested in applied linear algebra may want to examine this book for useful ideas. I can see parts of this book used as enrichment for an undergraduate linear algebra class or as material for an undergraduate research project.

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.