I suppose by this wording you mean that the subspace is invariant under ?1. is an invariant subspace relative to .
Let . Then for some .
Hence and so ; is therefore invariant under .
The condition on the column rank implies that if is non-zero then is also non-zero.2. If is of column rank , the every eigenvalue of is an eigenvalue of .
Now let be an eigenvalue of with non-zero eigenvector .
Let so that, by the remark above, is non-zero. Then
Since is non-zero, clearly is an eigenvalue of .