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Thread: Matrix problem

  1. #1
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    Matrix problem

    If $\displaystyle A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, find $\displaystyle c,d$

    so that $\displaystyle (c I + d A)^2 = A$
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  2. #2
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    Quote Originally Posted by flintstone View Post
    If $\displaystyle A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, find $\displaystyle c,d$

    so that $\displaystyle (c I + d A)^2 = A$

    I will assume that c and d are scalar constants, while I is the indentitiy matrix. Let's start out by doing some of the small arithmetic:

    $\displaystyle cI = \begin{pmatrix}c & 0 \\ 0 & c\end{pmatrix}$

    $\displaystyle dA = \begin{pmatrix}0 & d \\ -d & 0\end{pmatrix}$


    $\displaystyle
    cI + dA = \begin{pmatrix}c & d \\ -d & c\end{pmatrix}
    $



    $\displaystyle
    (cI + dA)^2 = \begin{pmatrix}c & d \\ -d & c\end{pmatrix}
    \begin{pmatrix}c & d \\ -d & c\end{pmatrix} =
    \begin{pmatrix}c^2-d^2 & 2cd \\ -2cd & c^2-d^2\end{pmatrix}
    $


    Now you should be able to set your c's and d's so that the matrix looks all like original A again.

    -Andy
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