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Thread: Algebra, Problems For Fun (17)

  1. #1
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    Algebra, Problems For Fun (17)

    A square matrix $\displaystyle J$ with entries from $\displaystyle \mathbb{C}$ is called a Jordan matrix if $\displaystyle J=\begin{pmatrix}\lambda & 1 & 0 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & 0 & \cdots & 0 \\ . & . & . & . & \cdots & . \\ . & . & . & . & \cdots & . \\ . & . & . & . & \cdots & . \\ 0 & 0 & 0 & 0 & \cdots & \lambda \end{pmatrix},$ for some $\displaystyle \lambda \in \mathbb{C}.$ Show that if $\displaystyle |\lambda| < 1,$ then $\displaystyle \lim_{n\to\infty}J^n = \bold{0}.$


    Remark: Using Jordan normal form the above result is easily extended to any matrix $\displaystyle A$ with entries from $\displaystyle \mathbb{C}$ and with this property that $\displaystyle |\lambda| < 1,$ for any eigenvalue $\displaystyle \lambda$ of $\displaystyle A.$
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  2. #2
    Super Member PaulRS's Avatar
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    We write: $\displaystyle
    J = \lambda \cdot {\text{Id}} + M
    $ where $\displaystyle
    M = \left( {\begin{array}{*{20}c}
    0 & 1 & 0 & \cdots & 0 \\
    {} & 0 & {} & {} & \vdots \\
    \vdots & {} & \ddots & {} & 0 \\
    {} & {} & {} & 0 & 1 \\
    0 & {} & \cdots & {} & 0 \\

    \end{array} } \right)
    $

    Now note that M represents the following digraph: 1->2->3->...->n . When we exponentiate M (remember that $\displaystyle
    \left[ {M^m } \right]_{i,j}
    $ is the number of paths of lenght m from i to j ) we get: $\displaystyle
    \left[ {M^m } \right]_{i,j} = \left\{ \begin{gathered}
    1{\text{ if }}i + m \leqslant {\text{dimension and }}j = i + m \hfill \\
    0{\text{ otherwise}} \hfill \\
    \end{gathered} \right.
    $ (1) (when I say dimension I am talking about the dimension of our matrices) - From here it is clear that $\displaystyle
    M^{{\text{dimension}}} =\bold{0}
    $. (2)

    Where does this come in handy? Well: $\displaystyle
    J^n = \left( {\lambda \cdot {\text{Id}} + M} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}\cdot M^k \cdot \lambda ^{n - k} }
    $ ( $\displaystyle M^0=\text{Id}$ )

    Consider $\displaystyle n\geq \text{dimension}-1$ then - by (2) - $\displaystyle
    J^n = \sum\limits_{k = 0}^{{\text{dimension}-1}} {\binom{n}{k}\cdot M^k \cdot \lambda ^{n - k} }
    $

    Then $\displaystyle
    \left[ {J^n } \right]_{i,j} = \sum\limits_{k = 0}^{{\text{dimension}-1}} {\binom{n}{k}\left[ {M^k } \right]_{i,j} \cdot \lambda ^{n - k} }
    $ but, look back at (1) if our entry here is not 0 already, it is only one of the terms of the sum , i.e. $\displaystyle
    \left[ {J^n } \right]_{i,j} = \binom{n}{l} \lambda ^{n - l}
    $ for some $\displaystyle 0\leq{l}\leq{\text{dimension}-1}$ and this clearly tends to 0 since $\displaystyle l$ is fixed ; $\displaystyle \binom{n}{l}\approx{\tfrac{n^l}{l!}}$ and $\displaystyle |\lambda|<1$
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  3. #3
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    Quote Originally Posted by PaulRS View Post
    We write: $\displaystyle
    J = \lambda \cdot {\text{Id}} + M
    $ where $\displaystyle
    M = \left( {\begin{array}{*{20}c}
    0 & 1 & 0 & \cdots & 0 \\
    {} & 0 & {} & {} & \vdots \\
    \vdots & {} & \ddots & {} & 0 \\
    {} & {} & {} & 0 & 1 \\
    0 & {} & \cdots & {} & 0 \\

    \end{array} } \right)
    $

    Now note that M represents the following digraph: 1->2->3->...->n . When we exponentiate M (remember that $\displaystyle
    \left[ {M^m } \right]_{i,j}
    $ is the number of paths of lenght m from i to j ) we get: $\displaystyle
    \left[ {M^m } \right]_{i,j} = \left\{ \begin{gathered}
    1{\text{ if }}i + m \leqslant {\text{dimension and }}j = i + m \hfill \\
    0{\text{ otherwise}} \hfill \\
    \end{gathered} \right.
    $ (1) (when I say dimension I am talking about the dimension of our matrices) - From here it is clear that $\displaystyle
    M^{{\text{dimension}}} =\bold{0}
    $. (2)

    Where does this come in handy? Well: $\displaystyle
    J^n = \left( {\lambda \cdot {\text{Id}} + M} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}\cdot M^k \cdot \lambda ^{n - k} }
    $ ( $\displaystyle M^0=\text{Id}$ )

    Consider $\displaystyle n\geq \text{dimension}-1$ then - by (2) - $\displaystyle
    J^n = \sum\limits_{k = 0}^{{\text{dimension}-1}} {\binom{n}{k}\cdot M^k \cdot \lambda ^{n - k} }
    $

    Then $\displaystyle
    \left[ {J^n } \right]_{i,j} = \sum\limits_{k = 0}^{{\text{dimension}-1}} {\binom{n}{k}\left[ {M^k } \right]_{i,j} \cdot \lambda ^{n - k} }
    $ but, look back at (1) if our entry here is not 0 already, it is only one of the terms of the sum , i.e. $\displaystyle
    \left[ {J^n } \right]_{i,j} = \binom{n}{l} \lambda ^{n - l}
    $ for some $\displaystyle 0\leq{l}\leq{\text{dimension}-1}$ and this clearly tends to 0 since $\displaystyle l$ is fixed ; $\displaystyle \binom{n}{l}\approx{\tfrac{n^l}{l!}}$ and $\displaystyle |\lambda|<1$
    beautiful! now suppose that $\displaystyle A$ is a square matrix with entries in $\displaystyle \mathbb{C}.$ let $\displaystyle \lambda_1, \cdots , \lambda_r$ be all distinct eigenvalues of $\displaystyle A.$ then we have: $\displaystyle A=P^{-1}BP,$ where $\displaystyle P$ is some invertible matrix and

    $\displaystyle B=\begin{pmatrix}J_1 & & & & & & \\ & J_2 & & & & & \\ & & . & & & & \\ & & & . & & & \\ & & & & . & & \\ & & & & & & J_r
    \end{pmatrix}, $ where each $\displaystyle J_k$ is a Jordan matrix with $\displaystyle \lambda_k$ on the diagonal. anywhere else in $\displaystyle B$ is 0. therefore: $\displaystyle A^n=P^{-1}B^nP=P^{-1}\begin{pmatrix}J_1^n & & & & & & \\ & J_2^n & & & & & \\ & & . & & & & \\ & & & . & & & \\ & & & & . & & \\ & & & & & & J_r^n
    \end{pmatrix}P$

    and thus if $\displaystyle |\lambda_k|<1$ for all $\displaystyle k,$ then $\displaystyle \lim_{n\to\infty}A^n=\bold{0}.$ this is an important result in applied and pure linear algebra and a key to the solution of skeboy's question in http://www.mathhelpforum.com/math-he...-matrices.html.

    skeboy doesn't seem to be looking for a rigorous proof of his question, which, by the way, is a part of the beautiful Perron-Frobenius theorem. i'll post my solution to his problem later.
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