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Define $$M(n,M\times N)=\{X\in \mathbb{R}^{M\times N} \mid \operatorname{rank} X = n\}$$ We know:

  1. $M(n,M\times N)$ is a smooth and connected manifold.
  2. The tangent space of $M(n,M\times N)$ at an element $X$ is $$T_X M(n,M\times N) = \{AX + XB \mid A\in \mathbb{R}^{M\times M}, B\in \mathbb{R}^{N\times N}\}$$

My question is the following lemma:

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Does the green part means $$A(t)X(t) + X(t)B(t)$$ is of rank $n$?

If yes, why? I cannot follow this.

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    It doesn't mean that the product $AX + XB$ is rank $n$, only that it defines a vector field. In order for the rank to be reduced, two vectors would need to coalesce under the action of $X(t)$ as $t$ increases. For the rank to increase we would need $u$ such that $X(0) u = 0$, but for some $t\geq 0$, $X(t)u \neq 0$. Of course, both of these situations would violate the basic existence and uniqueness theory of ODEs. – 2017-02-25
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    @Titus I do not quite understand "For the rank to increase..." what is that? And could you please explain both cases violating existence and uniqueness theory of ODEs more? Thanks! – 2017-02-25
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    My other comment was just me thinking out loud, and might not be very helpful. The argument only needs $A(t)X(t) + X(t)B(t)$ to be in the tangent space, since integrating along tangent vectors cannot lead off of the manifold. A low-dimensional analog is if you have any continuous vector field that takes values in $TS^2$, integrating along the field creates a path on $S^2$ itself. – 2017-02-25

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