Partition of a matrix

Task number: 2419

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

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

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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