Let A be an n x n matrix with the property that the intersection of its column space and its null space is the 0 vector. That is: nullA(intersection)colA={0}. Show that A and A^2 have the same rank........(Headbang)

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- April 20th 2008, 03:48 AMflawlessThis is tricky..Linear algebra again
Let A be an n x n matrix with the property that the intersection of its column space and its null space is the 0 vector. That is: nullA(intersection)colA={0}. Show that A and A^2 have the same rank........(Headbang)

- April 20th 2008, 05:25 AMflyingsquirrel
Hi

First, you can show that , which might be useful. (rank-nullity theorem)

Then, showing that and share the same rank is equivalent to proving that (rank-nullity theorem again) Here, we are in a special case : as , (find a proof :D) you can try to show that which will give you and the equality of the dimensions.

For , you might take which can be decomposed as with and (because ) and develop