We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.

中文翻译：

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关于矩阵与其转置的快速乘法

我们提出了一种非交换算法，用于在 C 或任何有限域上使用 5 个块乘积（3 个递归调用和 2 个通用乘积）将 2x2 块矩阵与其转置相乘。 我们对双线性形式的空间使用几何考虑描述 2x2 矩阵乘积以获得该算法，我们展示了如何减少所涉及的加法次数。 任意维度的结果算法是矩阵乘法通过其转置到一般矩阵乘积的减少，通过先前已知的常数因子改进减少.最后，我们提出了具有低内存占用的计划，以支持有限域上的快速且内存高效的实际实现。总而言之，我们展示了如何在 LDLT 分解中使用我们的结果。