## Partition of a matrix

Split the matrix

$$\begin{pmatrix} 0&1&2&3\\ 3&2&1&4\\ 4&3&0&1\\ 3&0&-1&2\\ \end{pmatrix}$$

into a sum of two matrices, where one is symmetric (i.e. $$\mathbf A=\mathbf A^T$$) and the other antisymmetric (i.e. $$\mathbf A=-\mathbf A^T$$).

• #### Resolution

The symmetric part is the elementwise average, while the antisymmetric is the difference from the average.

• #### Result

$$\begin{pmatrix} 0&1&2&3\\ 3&2&1&4\\ 4&3&0&1\\ 3&0&-1&2\\ \end{pmatrix} = \begin{pmatrix} 0&2&3&3\\ 2&2&2&2\\ 3&2&0&0\\ 3&2&0&2\\ \end{pmatrix} + \begin{pmatrix} 0&-1&-1&0\\ 1&0&-1&2\\ 1&1&0&1\\ 0&-2&-1&0\\ \end{pmatrix}$$