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Math Help - Proving a Matrix Nilpotent

  1. #1
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    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
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  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
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    For part 1 merely apply the principle given to the matrix. Ne_1 is, by the principle, the first column of your matrix. This column is all zeros. Similarly, the first column is e_1, etc.

    For part 2, apply part 1. N^2e_i = N(Ne_i), and we know what Ne_i is. Part 3 immediately follows.

    For part 4, note that N^ie_j is the e_j^{\text{th}} column of N^i. For example, we wish to show that all the columns of N^4 are zero but N^4e_j = e_{j-4} (just adding a formula to part 2 - N^ie_j = e_{i-j}) and we are done as j \leq 4 and e_{0} is the zero vector.
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