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Matrix Proofs
Posted:
May 22, 2005 8:50 PM


Hi group,
I need help with this proof relating to strictly uppertriangular matrices.
Let A be an n x n strictly upper triangular matrix. Then the (i,jth entry of AA = A^2 is 0 if i >= j  1.
Pf: Let B = A^2. The (i,j)th entry of B is given by
b_{ij} = \sum_{k=1}^{n} a_{ik} a_{kj}.
If k >= j, a_{kj} = 0. If i >= k, a_{ik} = 0. If i >= j  1, then there is no k s.t. i < k < j. Therefore
b_ij = \sum_{i<k, k<j} a_{ik} b_{kj} ==> b_{ij} = 0.
Also, if anyone has any clues on these related problems, it would be greatly appreciated.
Suppose p is a given integer satisfying 1 <= p <= n 1 and that the entries b_{kj} of an n x n matrix B satisfy b_{kj} = 0 for k >= j  p. Show that the (i,j)th entry of the product AB is zero if i >= j  (p+1). Deduce from the previous result that A^n = 0.
James



