"From too much study, and from extreme passion, cometh madnesse."

There is relatively less published pertaining to matrix-valued convex sets and functions.

A partial remedy for venturing into hyperdimensional representations, such as the cubix or quartix, is to first vectorize matrices.  This device gives rise to the Kronecker product of matrices; a.k.a, direct product or tensor product.

One advantage to vectorization is existence of a traditional two-dimensional matrix representation for the second-order gradient of a real function with respect to a vectorized matrix.  To disadvantage is a large new but known set of algebraic rules
and the fact that its mere use does not generally guarantee two-dimensional matrix representation of gradients.

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