Strassen algorithm

Strassen algorithm for a symmetric matrix:

For a square matrix [math]\displaystyle{ A \in \mathbb{C}^{n \times n} }[/math] where [math]\displaystyle{ 2^k := n, k \in \mathbb{N}^* }[/math], the runtime [math]\displaystyle{ T_q(n) }[/math] of the Strassen algorithm for the matrix product [math]\displaystyle{ AA^T }[/math] is about [math]\displaystyle{ 2/3 }[/math] that of the original algorithm in [math]\displaystyle{ \mathcal{O}(n^{(_2 7)}) }[/math].

Proof: For [math]\displaystyle{ A := \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} }[/math], it holds that [math]\displaystyle{ A^TA = \begin{pmatrix} A_{11}A_{11}^T+A_{12}A_{12}^T & A_{11}A_{21}^T+A_{12}A_{22}^T \\ A_{21}A_{11}^T+A_{22}A_{12}^T & A_{21}A_{21}^T+A_{22}A_{22}^T \end{pmatrix} }[/math].

The rate of usual matrix multiplications (top right) is [math]\displaystyle{ 2/3.\square }[/math]

See also

List of mathematical symbols