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How to write matrices as tensors

  1. Jun 1, 2005 #1
    I have some simple questions on how to write matrices as tensors.

    1.
    [tex]
    \left(\begin{array}{cc}a_1\\a_2\end{array}\right)+
    \left(\begin{array}{cc}b_1\\b_2\end{array}\right)=
    \left(\begin{array}{cc}c_1\\c_2\end{array}\right)
    [/tex]

    is this equivalent to

    [tex]A^j + B^j = C^j[/tex]

    with j = 1,2.

    2.

    [tex]
    1/2\left(\begin{array}{cc}\dot{x}_1 & \dot{x}_2\end{array}\right)
    \left(\begin{array}{cc}3m & m\\-m & 3m\end{array}\right)
    \left(\begin{array}{cc}\dot{x}_1 \\\dot{x}_2\end{array}\right)
    [/tex]

    is this equivalent to

    [tex]
    1/2\dot{x}^{\mu}M_{\mu\nu}\dot{x}^{\nu}
    [/tex]

    [tex]
    \mu,\nu=1,2
    [/tex]

    and

    [tex]
    M_{11}=3m,...,M_{22}=3m
    [/tex]
     
  2. jcsd
  3. Jun 1, 2005 #2

    dextercioby

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    Science Advisor
    Homework Helper

    Yes,it's correct in the first case.In the second,u may put it

    [tex] \frac{1}{2}\dot{x}_{\mu}M^{\mu}{}_{\nu} \dot{x}^{\nu} [/tex]

    Daniel.
     
  4. Jun 1, 2005 #3
    thanks.

    Is my 2 wrong?
    Im struggling with the contravariant and covariant indicies.
    is it because a row vector is a covariant vector and the column vector is a contravariant vector you write it like that...but that cant be right.


    another question:
    If you have an expression like

    [tex]
    A^{ijk}B_k
    [/tex]

    i,j,k = 1,2

    this is equivalent to 4 expressions

    [tex]
    A^{111}B_1 + A^{112}B_2
    [/tex]

    [tex]
    A^{121}B_1 + A^{122}B_2
    [/tex]

    [tex]
    A^{211}B_1 + A^{212}B_2
    [/tex]

    [tex]
    A^{221}B_1 + A^{222}B_2
    [/tex]
     
  5. Jun 1, 2005 #4

    dextercioby

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    Science Advisor
    Homework Helper

    Nope,it's just

    [tex] A^{ij1}B_{1}+A^{ij2}B_{2} [/tex]

    ,that is a second rank double contravariant tensor with 4 components,the ones you have written.

    Daniel.
     
    Last edited: Jun 1, 2005
  6. Jun 1, 2005 #5
    ok...ty.

    2 again.

    if you have a tensor

    [tex]1/2\dot{x}^{\mu}M_{\mu\nu}\dot{x}^{\nu}[/tex]

    [tex]\mu,\nu=1,2[/tex]

    [tex]M_{11}=3m,...,M_{22}=3m[/tex]

    and write it as a matrix you _dont_ get

    [tex]1/2\left(\begin{array}{cc}\dot{x}_1 & \dot{x}_2\end{array}\right)\left(\begin{array}{cc}3m & m\\-m & 3m\end{array}\right)\left(\begin{array}{cc}\dot{x}_1 \\\dot{x}_2\end{array}\right)[/tex]

    ?
     
  7. Jun 1, 2005 #6

    dextercioby

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    Science Advisor
    Homework Helper

    Nope.You can't put that expression in matrix form.

    Daniel.
     
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