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I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices. For instance, consider the following:

Problem: Let $A$ be a $k\times l$ matrix, and $B$ be a $l \times k$ matrix, and $X$ be a $l\times l$ matrix. Suppose $X$ is nilpotent: $X^n = 0$. Find the complete set of constraints for the matrices $Y_j$:

\begin{equation} Y_j = A X^j B, \ \ \ j = 0,1,\dots, n-1 \end{equation}

Naively, one can write down each entry in $Y_j$ as independent variable and perform elimination theory to get constraints. However:

  1. Is it always true that those constraints expressed in terms of components can actually be packaged into matrix equations?

  2. Is there a faster way that one can deal with matrix directly to perform elimination, without having to rely on the naive method described above?

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  • $\begingroup$ I guess there is a way using matrix resolvent. But I have not figured out how. $\endgroup$
    – Kevin Ye
    Dec 30, 2015 at 16:31

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