We investigate the expressive power of $\mathsf{MATLANG}$, a formal language
for matrix manipulation based on common matrix operations and linear algebra.
The language can be extended with the operation $\mathsf{inv}$ of inverting a
matrix. In $\mathsf{MATLANG}+\mathsf{inv}$ 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 $\mathsf{eigen}$ for diagonalizing a matrix, which is
defined so that different eigenvectors returned for a same eigenvalue are
orthogonal. We show that $\mathsf{inv}$ can be expressed in
$\mathsf{MATLANG}+\mathsf{eigen}$. We put forward the open question whether
there are boolean queries about matrices, or generic queries about graphs,
expressible in $\mathsf{MATLANG} + \mathsf{eigen}$ but not in
$\mathsf{MATLANG}+\mathsf{inv}$. The evaluation problem for $\mathsf{MATLANG} +
\mathsf{eigen}$ is shown to be complete for the complexity class $\exists
\mathbf{R}$.

探究了基于常见矩阵操作和线性代数的矩阵操作形式语言 MA​TLANG 的表达能力，可以用矩阵求逆操作计算有向图的传递闭包，但不可能未经求逆操作就实现。