Some complete calculi for matrices

A matrix calculus is introduced with the intention of developing data structures suitable for a high level algorithmic language for mathematical programming. The paper investigates how the special structure of matrices can be described and utilized for efficient computing by saving memory space and superfluous operations. Sequences of matrices (and sequences of sequences of matrices) are considered, and matrix operators are extended to sequence operators and cumulative operators. Algorithms are given which use symbol manipulation of matrix expressions so as to find the forms best suited for computation. These forms are called normal forms. Several completeness results are obtained in the sense that for each expression an equivalent expression in normal form can be found within a specified calculus.

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