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Thread: Matrix and Vector Magnitude

  1. #1
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    Matrix and Vector Magnitude

    Given matrix M and vector X where:
    $\displaystyle

    \[M =
    \left( {\begin{array}{cc}
    cos\theta & sin\theta \\
    -sin\theta & cos\theta
    \end{array} } \right) \]

    \textit{and}
    X =
    \left( {\begin{array}{cc}
    x \\
    y
    \end{array} } \right)
    $
    Compute: $\displaystyle
    \frac { \left|{MX}\right|}{\left|{X}\right|}
    $
    So I found that the result is 1.
    The question is to explain the result and describe the action of M geometrically.
    I think the result means that
    $\displaystyle
    \[M =
    \left( {\begin{array}{cc}
    1 & 0 \\
    0 & 1
    \end{array} } \right) \]
    $
    which, geometrically, doesn't change the length of vector X. Would that be correct?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    $\displaystyle M=\begin{bmatrix}{\;\;\;\cos \theta}&{\sin \theta}\\{-\sin \theta}&{\cos \theta}\end{bmatrix}=\begin{bmatrix}{\cos (-\theta)}&{-\sin (-\theta)}\\{\sin (-\theta)}&{\;\;\;\cos (-\theta)}\end{bmatrix}$

    represents a rotation around $\displaystyle (0,0)$ and angle $\displaystyle -\theta$ so ,

    $\displaystyle \left\|{MX}\right\|= \left\|{X}\right\|\; \forall{X}\in{\mathbb{R}^2} $


    Fernando Revilla
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  3. #3
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    Another way of seeing that is to note that the determinant is 1:
    $\displaystyle \left|\begin{array}{cc}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{array}\right|= (cos(\theta))(cos(\theta))(- (sin(\theta))(-sin(\theta))= cos^2(\theta)+ sin^2(\theta)= 1$
    and so the length of a vector is not changed.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    But the length does not change by the fact of being $\displaystyle M$ orthogonal (in this case a rotation) and not by the fact of being $\displaystyle \det M=1$ . For example:

    $\displaystyle M=\begin{bmatrix}{1}&{1}\\{0}&{1}\end{bmatrix}$

    has determinant $\displaystyle 1$ , however

    $\displaystyle \left\|{(0,1)^t}\right\|=1\neq \sqrt{2}=\left\|{M(0,1)^t}\right\|$


    Fernando Revilla
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  5. #5
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    Ouch! Good point!
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