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Thread: Matrix in partitioned form.

  1. #1
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    Matrix in partitioned form.

    Let $\displaystyle B$ be an $\displaystyle m * m$ matrix with inverse $\displaystyle B^-1$, $\displaystyle c$ an $\displaystyle m$-vector and $\displaystyle 0$ the $\displaystyle m$ dimensional null vector. $\displaystyle A$ is given in the following partitioned form...

    $\displaystyle A = \left[ \begin{array}{cccc} B & 0 \\ c^T & 1 \end{array} \right]$

    Determine $\displaystyle A^{-1}$ symbolically in a partition form.

    For which I get the following...

    $\displaystyle A^{-1} = \left[ \begin{array}{cccc} d & e \\ f & g \end{array} \right]$

    $\displaystyle \left[ \begin{array}{cccc} B & 0 \\ c^T & 1 \end{array} \right] \left[ \begin{array}{cccc} d & e \\ f & g \end{array} \right] = \left[ \begin{array}{cccc} 1 & 0 \\ 0 & 1 \end{array} \right]$

    $\displaystyle Bd = 1$

    $\displaystyle d = B^{-1}$

    $\displaystyle Be = 0$

    Since B is non-singular, B does not equal 0...

    $\displaystyle e = 0$

    $\displaystyle c^Td + f = 0$

    $\displaystyle c^TB^{-1} + f = 0$

    $\displaystyle f = -c^TB^{-1}$

    $\displaystyle c^Te + g = 1$

    $\displaystyle g = 1$

    $\displaystyle A^{-1} = \left[ \begin{array}{cccc} B^{-1} & 0 \\ -c^TB^{-1} & 1 \end{array} \right]$

    Is this correct so far?

    The question then asks the dimension of $\displaystyle A$ and the dimensions of the submatrices in $\displaystyle A^{-1}$.

    Am I right in thinking the dimension of A is m+1, the dimension of $\displaystyle B^{-1}$ is m and the dimension of $\displaystyle -c^TB^{-1}$ is 1?

    Thanks for the help.
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  2. #2
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    Quote Originally Posted by sean.1986 View Post
    Let $\displaystyle B$ be an $\displaystyle m * m$ matrix with inverse $\displaystyle B^-1$, $\displaystyle c$ an $\displaystyle m$-vector and $\displaystyle 0$ the $\displaystyle m$ dimensional null vector. $\displaystyle A$ is given in the following partitioned form...

    $\displaystyle A = \left[ \begin{array}{cccc} B & 0 \\ c^T & 1 \end{array} \right]$

    Determine $\displaystyle A^{-1}$ symbolically in a partition form.

    For which I get the following...

    ... [SNIP] ...

    $\displaystyle A^{-1} = \left[ \begin{array}{cccc} B^{-1} & 0 \\ -c^TB^{-1} & 1 \end{array} \right]$

    Is this correct so far? Yes.

    The question then asks the dimension of $\displaystyle A$ and the dimensions of the submatrices in $\displaystyle A^{-1}$.

    Am I right in thinking the dimension of A is m+1, the dimension of $\displaystyle B^{-1}$ is m and the dimension of $\displaystyle -c^TB^{-1}$ is 1?
    A matrix is a two-dimensional structure. So the dimensions of A are that it is an $\displaystyle (m+1)\mathord\times(m+1)$ matrix; $\displaystyle B^{-1}$ is mm; and $\displaystyle -c^TB^{-1}$ is 1m (or the transpose of an m-vector). Also, the 0 is an m1 matrix (or an m-vector), and the 1 is a 1×1 matrix.
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  3. #3
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    Ah that makes sense, thank you.
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