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Thread: Find a vector given a matrix

  1. #1
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    Find a vector given a matrix

    We did a problem in class I didn't understand.


    "Find a vector w so that:"
    [1 3]w = 2w
    [1 -1]

    I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...
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    Re: Find a vector given a matrix

    Quote Originally Posted by MrJank View Post
    We did a problem in class I didn't understand.
    "Find a vector w so that:"
    [1 3]w = 2w
    [1 -1]
    Use \[w = \left[ {\begin{array}{*{20}{c}}
    0 \\
    0
    \end{array}} \right]\]
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  3. #3
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    Re: Find a vector given a matrix

    Quote Originally Posted by Plato View Post
    Use \[w = \left[ {\begin{array}{*{20}{c}}
    0 \\
    0
    \end{array}} \right]\]
    The equality is a false statement. Any vector will be an nx1 matrix. We know n=2. The multiplication will yield a 1x1 matrix, which will never be equal to a scalar times a 2x1 matrix.
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  4. #4
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    Re: Find a vector given a matrix

    Quote Originally Posted by MrJank View Post
    We did a problem in class I didn't understand.


    "Find a vector w so that:"
    [1 3]w = 2w
    [1 -1]

    I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...
    Let w be the vector
    $\displaystyle w = \left [ \begin{matrix} a \\ b \end{matrix} \right ] $

    Then we need to find a, b such that the following is true:
    $\displaystyle \left [ \begin{matrix} 1 & 3 \\ 1 & -1 \end{matrix} \right ] ~ \left [ \begin{matrix} a \\ b \end{matrix} \right ] = 2 \left [ \begin{matrix} a \\ b \end{matrix} \right ] $

    Multiplying this out we get the two equations
    $\displaystyle \begin{matrix} a + 3b = 2a \\ a - b = 2b \end{matrix} $

    What can a and b be?

    -Dan
    Thanks from SlipEternal
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    Re: Find a vector given a matrix

    Quote Originally Posted by SlipEternal View Post
    The equality is a false statement. Any vector will be an nx1 matrix. We know n=2. The multiplication will yield a 1x1 matrix, which will never be equal to a scalar times a 2x1 matrix.
    ?????
    \[\left[ {\begin{array}{*{20}{c}}
    1&3 \\
    1&{ - 1}
    \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
    x \\
    y
    \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
    {x + 3y} \\
    {x - y}
    \end{array}} \right]

    \\\\\left[ {\begin{array}{*{20}{c}}
    1&3 \\
    1&{ - 1}
    \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
    0 \\
    0
    \end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}}
    {0} \\
    {0}
    \end{array}} \right]\]
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  6. #6
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    Re: Find a vector given a matrix

    It appears that the solution is any vector [w1, w2] where w1/w2=3.

    If I am not mistaken, this kind of solution often occurs in vibrations where multiple masses oscillate with any amplitude, however these amplitudes have fixed ratios (eigenvectors). Been a while though, I may be wrong.
    Last edited by troymius; Aug 26th 2018 at 04:06 PM.
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  7. #7
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    Re: Find a vector given a matrix

    Quote Originally Posted by Plato View Post
    ?????
    \[\left[ {\begin{array}{*{20}{c}}
    1&3 \\
    1&{ - 1}
    \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
    x \\
    y
    \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
    {x + 3y} \\
    {x - y}
    \end{array}} \right]

    \\\\\left[ {\begin{array}{*{20}{c}}
    1&3 \\
    1&{ - 1}
    \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
    0 \\
    0
    \end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}}
    {0} \\
    {0}
    \end{array}} \right]\]
    I thought the matrix was just the vector [1 3]. I didn't realize the [1 -1] was part of it.
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  8. #8
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    Re: Find a vector given a matrix

    Quote Originally Posted by MrJank View Post
    We did a problem in class I didn't understand.


    "Find a vector w so that:"
    [1 3]w = 2w
    [1 -1]

    I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...
    If it only equaled two it would be impossible! The left side is a vector while "2" is not.
    Write [tex]w= \begin{bmatrix} x \\ y \end{bmatrix}[tex]. The equation becomes
    $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x+ 3y \\ x- y \end{bmatrix}= 2\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2x \\ 2y \end{bmatrix}$.

    That is the same as the two equations x+ 3y= 2x and x- y= 2y. It should be easy to see that the only solution is x= y= 0.

    A little more "sophisticated" would be to write the equation as $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$.

    Then $\displaystyle \begin{bmatrix}1 - 2 & 3- 0 \\ 1- 0 & -1- 2\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}-1 & 3 \\ 1 & -3 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.

    Now, as long as that matrix is invertible, we could multiply both sides by its inverse and get, since the product of any matrix with the 0 vector is the 0 vector, $\displaystyle \begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.
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    Re: Find a vector given a matrix

    Quote Originally Posted by HallsofIvy View Post
    If it only equaled two it would be impossible! The left side is a vector while "2" is not.
    Write [tex]w= \begin{bmatrix} x \\ y \end{bmatrix}[tex]. The equation becomes
    $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x+ 3y \\ x- y \end{bmatrix}= 2\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2x \\ 2y \end{bmatrix}$.

    That is the same as the two equations x+ 3y= 2x and x- y= 2y. It should be easy to see that the only solution is x= y= 0.

    A little more "sophisticated" would be to write the equation as $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$.

    Then $\displaystyle \begin{bmatrix}1 - 2 & 3- 0 \\ 1- 0 & -1- 2\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}-1 & 3 \\ 1 & -3 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.

    Now, as long as that matrix is invertible, we could multiply both sides by its inverse and get, since the product of any matrix with the 0 vector is the 0 vector, $\displaystyle \begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.
    Actually, $w=c\begin{bmatrix}3\\1\end{bmatrix}$ where $c$ is any real number. It is finding an eigenvector for the matrix with eigenvalue 2.
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  10. #10
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    Re: Find a vector given a matrix

    Quote Originally Posted by MrJank View Post
    "Find a vector w so that:"
    [1 3]w = 2w
    [1 -1]...
    Quote Originally Posted by SlipEternal View Post
    Actually, $w=c\begin{bmatrix}3\\1\end{bmatrix}$ where $c$ is any real number. It is finding an eigenvector for the matrix with eigenvalue 2.
    Actually this is a poorly formed question and even more poorly presented one.
    It does say find a vector...
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  11. #11
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    Re: Find a vector given a matrix

    Quote Originally Posted by Plato View Post
    Actually this is a poorly formed question and even more poorly presented one.
    It does say find a vector...
    I agree. My point was not that $x=y=0$ is wrong. I was pointing out that it is not the only solution.
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