Solving the problem of simultaneous diagonalisation via congruence
Miguel D. Bustamante, P. Mellon, Maria Victoria Velasco · arXiv (Cornell University) · 2019
We provide a solution to the so-called $SD$ $problem$, that is, the problem of simultaneous $diagonalisation$ $via$ $congruence$ of a given set of $m$ symmetric $n\times n$ matrices $\{A_{1},\ldots,A_{m}\}$, by showing that it can be reduced to a lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous $diagonalisation$ $via$ $similarity$ of a new set of matrices. We provide a concrete algorithm to determine whether or not a set of matrices is simultaneously diagonalisable by congruence. This solves a long standing problem in the complex case. The SD problem has many applications in signal separation and optimisation.