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 .