We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in positive characteristic. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to alternatively compute the multiplication taking advantage of the structure of the matrix-polynomial arithmetic involved. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field.

中文翻译：

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一些快速算法将矩阵与其伴随相乘

我们提出了一种非交换算法，用于将2 x 2块矩阵乘以矩阵环反同态定义的伴随关系。该算法在C上或以正特性使用5个块积（3个递归调用和2个通用积）。所得到的用于任意尺寸的算法是通过将矩阵的相乘减少到与普通矩阵乘积的相乘来减少矩阵，将其乘以以前已知的减少常数。我们还证明了，仅使用四个乘积和其中一个乘积的双线性形式并没有派生出算法。其次，我们针对复数场和四元数给出了新颖的专用算法，以利用所涉及的矩阵多项式算法的结构来交替计算乘法。然后，我们分析两种策略各自的优势范围。最后，我们提出了具有低内存占用量的计划，这些计划支持在主要领域上实现快速且内存高效的实际实现。