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:

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

If you now look at the main diagonal's general entry of the above, you get

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...

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