Proving a Matrix Nilpotent

A matrix is called nilpotent if there is some positive integer d such that N^d =O (the zero matrix).

The smalles value of d which fullfills this is called degree of nilpotency.

Consider the following matrix:

N = 0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

Consider the following principle:

Let e1, e2...en be the collumns of the n x n identitiy matrix I. Then for any n x n matrix A, the product Aej is exactly the jth collumn of A.

Now for the proof.

(1) Explain why Ne1 = 0 (a collumn matrix of 0s)

Ne2 = e1

Ne3 = e2

Ne4 = e4

(2) Use the four equations of part (1) to prove that

N^2e1 = 0 ( collumn matrix of 0s).

N^2e2 = 0 " "

N^2e3 = e1

N^2e4 = e2

(3) By this similar reasoning, find a set of four equations that characterizes N^3 and likewise N^4.

(4) Interpret the results of part (2) and (3) with the principle mentioned above to show that

N^2 =

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

N^3 =

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

N^4 =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(5) Finally prove that if Ni nilpotent of degree 4, then

N^d= N^(d+1) = N^(d+2)= ... = O (the zero matrix).

Thanks if anyone can help me with this