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 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 9R.

中文翻译：

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马特朗

我们研究了 MATLANG 的表达能力，这是一种基于通用矩阵运算和线性代数的矩阵操作形式语言。该语言可以通过用于反转矩阵的操作 inv 进行扩展。在 MATLANG + inv 中，我们可以计算有向图的传递闭包，而我们表明，如果没有反转，这是不可能的。事实上，我们证明了基本语言可以在关系代数中用算术运算、分组和求和来模拟。我们还考虑了对矩阵进行对角化的操作特征。它被定义为对于每个特征值，返回一组正交特征向量，它们跨越该特征值的特征空间。我们证明 inv 可以用 MATLANG + eigen 表示。我们提出了开放性问题，是否存在关于矩阵的布尔查询或关于图的通用查询，可以在 MATLANG+eigen 中表达，但不能在 MATLANG+inv 中表达。最后，对于复杂度等级 9R，MATLANG + eigen 的评估问题被证明是完整的。