On fast multiplication of a matrix by its transpose

We present a non-commutative algorithm for the multiplication of a 2 × 2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any field of prime characteristic. We use geometric considerations on the space of bilinear forms describing 2 × 2 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 prime field. To conclude, we show how to use our result in L · D · LT factorization.