On the Expressive Power of Query Languages for Matrices

Abstract

We investigate the expressive power of MATLANG, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation inv for inverting a matrix. In MATLANG + inv, we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed, we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation eigen for diagonalizing a matrix. It is defined such that for each eigenvalue a set of mutually orthogonal eigenvectors is returned that span the eigenspace of that eigenvalue. We show that inv can be expressed in MATLANG + eigen. We put forward the open question whether there are Boolean queries about matrices, or generic queries about graphs, expressible in MATLANG + eigen but not in MATLANG + inv. Finally, the evaluation problem for MATLANG + eigen is shown to be complete for the complexity class ∃ R.