Using Neumann series to compute $T^{-1}$

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Solution 1

Writing $T = I_3-S$ for $$ S= -\begin{pmatrix} 0 & \tfrac{1}{2} & \tfrac{1}{3} \\ 0 & 0 & \tfrac{1}{2} \\ 0 & 0 & 0 \end{pmatrix}, $$ you don't want to use the Neumann series to compute $S^{-1} = (I_3-T)^{-1}$ in terms of $T$, but rather $T^{-1} = (I_3-S)^{-1}$ in terms of $S$, i.e., $$ T^{-1} = (I_3-S)^{-1} = \sum_{k=0}^\infty S^k; $$ to check that this series indeed converges and gives $T^{-1}$, you must show that $\|S\| < 1$ for the operator norm $$ \|S\| = \sup_{x \neq 0} \frac{\|Sx\|}{\|x\|} = \sup_{\|x\|=1}\|Sx\|. $$ Now, let $x = (a,b,c)^T \in \mathbb{R}^3$ with $\|x\|=1$, so that $1=\|x\| = |a|+|b|+|c|$. Then $$ \|Sx\| = \left\|-\begin{pmatrix} 0 & \tfrac{1}{2} & \tfrac{1}{3} \\ 0 & 0 & \tfrac{1}{2} \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix}a\\b\\c\end{pmatrix}\right\|\\ = \left\| -\begin{pmatrix}\tfrac{1}{2}b+\tfrac{1}{3}c \\ \tfrac{1}{2}c \\ 0 \end{pmatrix} \right\|\\ = \left|\tfrac{1}{2}b+\tfrac{1}{3}c\right|+\left|\tfrac{1}{2}c\right|\\ \leq \tfrac{1}{2}|b| + \tfrac{5}{6}|c|\\ \leq \tfrac{5}{6}(|a|+|b|+|c|)\\ = \tfrac{5}{6}, $$ so that $$ \|S\| = \sup_{\|x\|=1}\|Sx\| \leq \tfrac{5}{6} < 1, $$ as required for applicability of the Neumann series.

Now, you can compute away, though your life is made easy by the fact that $S$ is nilpotent -- indeed, $S^3 = 0$ -- so that your Neumann series for $T^{-1} = (I_3-S)^{-1}$ is in fact a finite sum.

Solution 2

I am not sure what the $\left\|\begin{pmatrix}a\\b\\c\end{pmatrix}\right\|$ bit is about. As long as $\|S\|<1$ for some matrix norm $\|\cdot\|$, the series $I+S+S^2+S^3+\cdots$ will converge to $(I-S)^{-1}$. Now $S= -\begin{pmatrix}0 &\frac12 &\frac13\\0 &0 &\frac12\\0 &0 &0\end{pmatrix}$. Its Frobenius norm is $\|S\|_F=\sqrt{\frac1{2^2}+\frac1{3^2}+\frac1{2^2}}=\sqrt{\frac12+\frac19}<1$. Hence $T^{-1}=(I-S)^{-1}=I+S+S^2+S^3+\cdots$. Since $S$ is a nilpotent $3\times3$ matrix, we must have $S^3=0$. Hence $T^{-1}=I+S+S^2$.

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Updated on August 01, 2022

Comments

  • User69127
    User69127 over 1 year

    Need help on how to show that $S$ satisfies the necessary condition for Neumann series. Here is what is given. $T\in B(X,X)$ where $X$ is a Banach space. Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ be a bounded linear operator with matrix representation

    $$ \begin{pmatrix} 1 & 1/2 & 1/3 \\ 0 & 1 & 1/2 \\ 0 & 0 & 1 \\ \end{pmatrix} $$

    Use the Neumann series to compute $T^{-1}$. The operator $T$ can be written as $I-S$. Show that S satisfies the necessary condition for the Neumann series.

    So here is my plan of attack: I know I have to use the norm

    $$\left\|\begin{pmatrix} a \\ b \\ c \\ \end{pmatrix}\right\| = |a|+|b|+|c|$$

    to bound $\|S\|$. I also know I have to use the the Neumann series

    $$(I-S)^{-1}=\sum_{j=0}^\infty S^j=I+S+S^2+\cdots$$

    But how can I get $T^{-1}$ from this formula? Any help will be appreciated!

    • Raskolnikov
      Raskolnikov over 10 years
      I think you turned things around, you should compute $T^{-1}=(I-S)^{-1}$ with the Neumann series.
    • User69127
      User69127 over 10 years
      I believe you're right let me edit.
    • Branimir Ćaćić
      Branimir Ćaćić over 10 years
      Also, is the $1$ in the $(2,1)$ entry really there, or is that a typo? Because if that $1$ really is there, then $S(1,0,0)^T = -(1,0,0)^T$, and hence $\|S\| \geq 1$, making use of the Neumann series impossible.
    • User69127
      User69127 over 10 years
      Typo again. You're good at this! Just edited
  • User69127
    User69127 over 10 years
    Definitely helpful thank you very much!
  • User69127
    User69127 over 10 years
    Short, concice, and very helpful thank you as well