看了一个帖子 https://math.stackexchange.com/questions/94287/full-rank-condition-for-product-of-two-matrices?rq=1
其中有一个回答是这样的:
Somewhat expanding the comments, one can say the following. The rank of the product of a m×n and a n×p matrix can never exceed n (nor can it exceed m or p, but that is obvious from the size of the product). This is because, if you want to find r (for rank) vectors in the source space (of dimension p) whose images by the product of the matrices are linearly independent, then their images by the first matrix applied (B) must also be linearly independent, and this can only be the case if the dimension if the intermediate space permits it: r≤n. So if n<min(m,p) then the product can never have full rank.
If min(m,p)≤n≤max(m,p) then the product will have full rank if both matrices in the product have full rank: depending on the relative size of m and p the product will then either be a product of two injective or of two surjective mappings, and this is again injective respectively surjective. This condition of full rank factors is not a necessary one though: if m≥n>p then A might have a nonzero kernel, as long as it forms a direct sum with the image of B, while if m<n≤p then the image of B need not be the whole space, as long as its sum with the kernel of A fills that space.
When n>max(m,p) all one can say in general, knowing just the ranks of A and B, is the usual necessary condition either that B be injective (if m≥p) or that A be surjective (if m≤p) in order for AB to be injective respectively surjective, that is, full rank. Again the precise condition for AB to be injective is kerA⊕imB (this gives "full rank" when m≥p), and for AB to be surjective is kerA+imB=Rn (this gives "full rank" when m≤p).
It may be interesting to note that one can define the rank of a m×p matrix C as the minimal value r such that there exists a decomposition C=AB with A of size m×r and B of size r×p. It does not immediately give a good method to compute the rank, but lets you understand immediately that you cannot have full rank in your question unless n≥min(m,p).
感觉很相关,但是很多地方看不太懂。
Marc van Leeuwen (https://math.stackexchange.com/users/18880/marc-van-leeuwen), Full-rank condition for product of two matrices, URL (version: 2013-12-26): https://math.stackexchange.com/q/189942