## Saturday, March 1, 2014

### Show that A^2 = 0 is possible but A’A = 0 is not possible (unless A= zero matrix).

Found a solution in Stackexchage and quoting it here:

http://math.stackexchange.com/questions/435880/if-aat-is-the-zero-matrix-then-a-is-the-zero-matrix

if we put A=(aij)1i,jn , then At=(bij) , with bij=aji , so by definition:
AAt=(k=1naikbkj)=(k=1naikajk)

If you now look at the main diagonal's general entry of the above, you get
k=1naikaik=k=1na2ik

So if AAt=0 then the above diagonal's entries are zero, but a sum of squared real numbers is zero iff each number is zero, so...