Sub-direct sums and positivity classes of matrices

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.