Sub-direct sums and positivity classes of matrices

AbstractIt 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 known that this statement remains true if positive semidefinite is replaced with: doubly nonnegative, completely positive, totally nonnegative, M-matrix and P-matrix, etc. For each of these classes we consider corresponding questions for a more general “sum” of two matrices, of which the direct sum and ordinary sum are special cases