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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?
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  • $\begingroup$ The isotropic-deviatoric splitting is common in the field of Continuum Mechanics $$\eqalign{ {\rm iso}(X) &= \frac{ {\rm tr}(X) } { {\rm tr}(I)}I \cr {\rm dev}(X) &= X - {\rm iso}(X)}$$The deviatoric part is traceless. $\endgroup$ – lynn Jan 10 '16 at 21:28

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