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Math Help - Calculating powers of a superdiagonal matrix

  1. #1
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    Calculating powers of a superdiagonal matrix

    Hello,

    while dealing with nonhomogeneous equations with constant coefficients I ecountered a following problem - I need to calculate powers of a given matrix (all powers up to n-1):

    \mathbb N^{n}_{n} \ni \mathbb M_{n}= \begin{bmatrix} 0&n-1&0&0&...&0&0&0&0\\0&0&n-2&0&...&0&0&0&0\\0&0&0&n-3&...&0&0&0&0\\...&...&...&...&...&...&...&...&...  \\0&0&0&0&...&0&3&0&0\\0&0&0&0&...&0&0&2&0\\0&0&0&  0&...&0&0&0&1\\0&0&0&0&...&0&0&0&0 \end{bmatrix}

    Is there an easy way to calculate it?
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  2. #2
    Super Member girdav's Avatar
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    Re: Calculating powers of a superdiagonal matrix

    We can show by induction on p that (M_n^p)_{i,j} = 0 if i\geq n-p or j\neq i+p and (M_n^p)_{i,i+p}=\prod_{j=0}^{p-1}(n-i-j).
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  3. #3
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    Re: Calculating powers of a superdiagonal matrix

    Quote Originally Posted by protaktyn View Post
    I need to calculate powers of a given matrix (all powers up to n-1):

    \mathbb N^{n}_{n} \ni \mathbb M_{n}= \begin{bmatrix} 0&n-1&0&0&...&0&0&0&0\\0&0&n-2&0&...&0&0&0&0\\0&0&0&n-3&...&0&0&0&0\\...&...&...&...&...&...&...&...&...  \\0&0&0&0&...&0&3&0&0\\0&0&0&0&...&0&0&2&0\\0&0&0&  0&...&0&0&0&1\\0&0&0&0&...&0&0&0&0 \end{bmatrix}

    Is there an easy way to calculate it?
    I guess the easiest way is just to do the matrix multiplication to find \mathbb M_{n}^2 and \mathbb M_{n}^3. You will then see the pattern for \mathbb M_{n}^k (and if you really want to prove it, you can use induction).

    In fact, the nonzero elements of \mathbb M_{n}^k all lie on a single diagonal, namely the diagonal k places above the main diagonal. The elements on this diagonal, reading from top left to bottom right, are

    (n-1)(n-2)\cdots(n-k),\ (n-2)(n-3)\cdots(n-k-1),\ \ldots,\ k!.

    That formula works for 1\leqslant k\leqslant n-1. Then \mathbb M_{n}^k=0 for k\geqslant n.

    Edit. Beaten to it by girdav!
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  4. #4
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    Re: Calculating powers of a superdiagonal matrix

    With "i" being the row, or the column index? Just asking because I've seen both ways of indexing...
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  5. #5
    Super Member girdav's Avatar
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    Re: Calculating powers of a superdiagonal matrix

    Quote Originally Posted by protaktyn View Post
    With "i" being the row, or the column index? Just asking because I've seen both ways of indexing...
    "i" is the column index.
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  6. #6
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    Re: Calculating powers of a superdiagonal matrix

    Thank you both.
    Somehow though I couldn't get used to the notation of {i,j} being the indexes of the column and row respectively, but I took my time and figured it out.
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