  We obtain a new inequality for arbitrary Hermitian matrices. We describe
particular linear maps called the matrix portrait of arbitrary NxN matrices.
The maps are obtained as analogs of partial tracing of density matrices of
multipartite qudit systems. The structure of the maps is inspired by "portrait"
map of the probability vectors corresponding to the action on the vectors by
stochastic matrices containing either unity or zero matrix elements. We obtain
new entropic inequalities for arbitrary qudit states including a single qudit
and an discuss entangled single qudit state. We consider in detail the examples
of N=3 and 4. Also we point out a possible use of entangled states of systems
without subsystems (e.g., a single qudit) as a resource for quantum
computations.
