On rank-critical matrix spaces

We then study rank-critical spaces in the context of compression and primitive matrix spaces. We first show that every rank-critical matrix space can be decomposed into a rank-critical compression matrix space and a rank-critical primitive matrix space. We then prove, using our necessary and sufficient condition, that the block-diagonal direct sum of two rank-critical matrix spaces is rank-critical if and only if both matrix spaces are primitive, when the field is large enough.