https://cxy0714.github.io/en/tags/symbolic-computation/Recent content in Symbolic Computation on Xingyu ChenHugoen-USFri, 17 Mar 2023 18:34:35 +0800https://cxy0714.github.io/en/2023/03/17/mathematica-matrix-symbolic/Fri, 17 Mar 2023 18:34:35 +0800https://cxy0714.github.io/en/2023/03/17/mathematica-matrix-symbolic/<h2 id="where-the-problem-came-from">Where the problem came from</h2> <p>Lately, in order to derive some formulas, I needed to verify the inner-product results of a number of high-dimensional matrices (more than twenty of them, each taking anywhere from one to eight hours by hand), and I needed a reliable tool to help me check my computations. So I thought of Mathematica.</p> <p>My computation was roughly this: take two $k^2\times k^5$ matrices $A,B$ and compute their inner product $Tr(A \cdot B^T)$. Using the cyclic property of $Tr$, this can be turned into a sort of &ldquo;inner product&rdquo; between a $1 \times k^2$ row vector, a $k^2\times k^5$ matrix and a $k^5\times 1$ column vector — that is, adding up a pile of ($k^2\times k^5$) numbers. But $A$ and $B$ are not scalar matrices: their entries are all symbolic, and the dimensions grow with the order of the formula.</p>