用同余法解决复对称矩阵同时对角化问题
Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence
摘要：
我们提供了一个解决方案，同时$diagonalization$ $via$ $congruence$的一组给定的$m$复杂的对称$n\times n$矩阵$\{A_{1},\ldots,A_{m}\}$的问题，通过表明它可以减少到一个可能的低维问题，该问题是rephrased在一个新的相关矩阵集的同时$diagonalization$ $via$ $similarity$的经典问题。我们提供了一个程序，以确定在有限的步骤是否一组矩阵是同时可对角化的合同。这解决了复杂情况下长期存在的问题。
英文摘要：
We provide a solution to the problem of simultaneous $diagonalization$ $via$ $congruence$ of a given set of $m$ complex symmetric $n\times n$ matrices $\{A_{1},\ldots,A_{m}\}$, by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous $diagonalization$ $via$ $similarity$ of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.
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