It is well known that a direct sum is positive semidefinite if and only if
each of the direct summands is positive semidefinite. In fact, it is also k
nown that this statement remains true if positive semidefinite is replaced
with: doubly nonnegative, completely positive, totally nonnegative, M-matri
x and P-matrix, etc. For each of these classes we consider corresponding qu
estions for a more general "sum" of two matrices, of which the direct sum a
nd ordinary sum are special cases. (C) 1999 Elsevier Science Inc. All right
s reserved.