Matrix splitting procedures - is there equivalent of Helmholtz decomposition?

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My post consists of two separate questions:

  • I am interested in different ways to split a matrix in a form:

    A = B + C

where both B and C would have some specific, useful properties. I am familiar with splitting the of the matrix in the symmetric and antisymmetric part. Are there other useful ways and does someone have a list of these in a nice, structured way?

  • second, is there an equivalent of Helmholtz decomposition for matrices, i.e. splitting of the matrix in such a way that one matrix corresponds to rotational, and other to solenoidal field?