What is the fundamental matrix solution?












0












$begingroup$


Let $A$ be a 3 by 3 matrix, such that $dot{x}=Ax$. I am trying to find the fundamental matrix solution. I know that I need to find the eigenvalues and eigenvectors of $A$ which I did but I am not sure what to do next. Does anyone know what to once you find the eigenvalues and eigenvectors?



Thanks in advance.






matrices ordinary-differential-equations systems-of-equations matrix-equations







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Do you mean the fundamental matrix of a system of differential equations?
    $endgroup$
    – cheesyfluff
    Jan 2 '16 at 16:00










  • $begingroup$
    @cheesyfluff, yes.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:16










  • $begingroup$
    Do you know about matrix exponentials?
    $endgroup$
    – Jack M
    Jan 2 '16 at 16:17










  • $begingroup$
    @JackM, Yes, I do.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:18
















0












$begingroup$


Let $A$ be a 3 by 3 matrix, such that $dot{x}=Ax$. I am trying to find the fundamental matrix solution. I know that I need to find the eigenvalues and eigenvectors of $A$ which I did but I am not sure what to do next. Does anyone know what to once you find the eigenvalues and eigenvectors?



Thanks in advance.






matrices ordinary-differential-equations systems-of-equations matrix-equations







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Do you mean the fundamental matrix of a system of differential equations?
    $endgroup$
    – cheesyfluff
    Jan 2 '16 at 16:00










  • $begingroup$
    @cheesyfluff, yes.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:16










  • $begingroup$
    Do you know about matrix exponentials?
    $endgroup$
    – Jack M
    Jan 2 '16 at 16:17










  • $begingroup$
    @JackM, Yes, I do.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:18














0












0








0





$begingroup$


Let $A$ be a 3 by 3 matrix, such that $dot{x}=Ax$. I am trying to find the fundamental matrix solution. I know that I need to find the eigenvalues and eigenvectors of $A$ which I did but I am not sure what to do next. Does anyone know what to once you find the eigenvalues and eigenvectors?



Thanks in advance.






matrices ordinary-differential-equations systems-of-equations matrix-equations







share|cite|improve this question











$endgroup$




Let $A$ be a 3 by 3 matrix, such that $dot{x}=Ax$. I am trying to find the fundamental matrix solution. I know that I need to find the eigenvalues and eigenvectors of $A$ which I did but I am not sure what to do next. Does anyone know what to once you find the eigenvalues and eigenvectors?



Thanks in advance.






matrices ordinary-differential-equations systems-of-equations matrix-equations




matrices ordinary-differential-equations systems-of-equations matrix-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 2 '16 at 16:57









cheesyfluff

1,482520




1,482520










asked Jan 2 '16 at 15:55









MaxMinMaxMin

11




11








  • 1




    $begingroup$
    Do you mean the fundamental matrix of a system of differential equations?
    $endgroup$
    – cheesyfluff
    Jan 2 '16 at 16:00










  • $begingroup$
    @cheesyfluff, yes.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:16










  • $begingroup$
    Do you know about matrix exponentials?
    $endgroup$
    – Jack M
    Jan 2 '16 at 16:17










  • $begingroup$
    @JackM, Yes, I do.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:18














  • 1




    $begingroup$
    Do you mean the fundamental matrix of a system of differential equations?
    $endgroup$
    – cheesyfluff
    Jan 2 '16 at 16:00










  • $begingroup$
    @cheesyfluff, yes.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:16










  • $begingroup$
    Do you know about matrix exponentials?
    $endgroup$
    – Jack M
    Jan 2 '16 at 16:17










  • $begingroup$
    @JackM, Yes, I do.
    $endgroup$
    – MaxMin
    Jan 2 '16 at 16:18








1




1




$begingroup$
Do you mean the fundamental matrix of a system of differential equations?
$endgroup$
– cheesyfluff
Jan 2 '16 at 16:00




$begingroup$
Do you mean the fundamental matrix of a system of differential equations?
$endgroup$
– cheesyfluff
Jan 2 '16 at 16:00












$begingroup$
@cheesyfluff, yes.
$endgroup$
– MaxMin
Jan 2 '16 at 16:16




$begingroup$
@cheesyfluff, yes.
$endgroup$
– MaxMin
Jan 2 '16 at 16:16












$begingroup$
Do you know about matrix exponentials?
$endgroup$
– Jack M
Jan 2 '16 at 16:17




$begingroup$
Do you know about matrix exponentials?
$endgroup$
– Jack M
Jan 2 '16 at 16:17












$begingroup$
@JackM, Yes, I do.
$endgroup$
– MaxMin
Jan 2 '16 at 16:18




$begingroup$
@JackM, Yes, I do.
$endgroup$
– MaxMin
Jan 2 '16 at 16:18










1 Answer
1






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oldest

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0












$begingroup$

Assuming you have three unique eigenvalues $(lambda_1,lambda_2,lambda_3)$ and eigenvectors $(pmb{xi}^{(1)},pmb{xi}^{(2)},pmb{xi}^{(3)})$ you should have three linearly independent solutions in the form
$$mathbf{x}_1(t)=pmb{xi}^{(1)} e^{lambda_1t}qquad
mathbf{x}_2(t)=pmb{xi}^{(2)} e^{lambda_2t}qquad
mathbf{x}_3(t)=pmb{xi}^{(3)} e^{lambda_3t}$$
Then your fundamental matrix should be
$$pmb{psi}(t)=left(
begin{array}{@{}ccc@{}}
mathbf{x}_1(t)&
mathbf{x}_2(t)&
mathbf{x}_3(t)
end{array}right)=left(
begin{array}{@{}ccc@{}}
xi_1^{(1)e^{lambda_1t}}&xi_1^{(2)e^{lambda_2t}}&xi_1^{(3)e^{lambda_3t}}\
xi_2^{(1)e^{lambda_1t}}&xi_2^{(2)e^{lambda_2t}}&xi_2^{(3)e^{lambda_3t}}\
xi_3^{(1)e^{lambda_1t}}&xi_3^{(2)e^{lambda_2t}}&xi_3^{(3)e^{lambda_3t}}\
end{array}
right)
$$
where $xi_n^{(m)}$ denotes the $n$th element of $pmb{xi}^{(m)}$. Note that the general solution of the differential equation is
$$mathbf{x}=pmb{psi}(t)mathbf{c}$$
where $mathbf{c}=(c_1,c_2,c_3)^intercal$ is a constant matrix.






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    0












    $begingroup$

    Assuming you have three unique eigenvalues $(lambda_1,lambda_2,lambda_3)$ and eigenvectors $(pmb{xi}^{(1)},pmb{xi}^{(2)},pmb{xi}^{(3)})$ you should have three linearly independent solutions in the form
    $$mathbf{x}_1(t)=pmb{xi}^{(1)} e^{lambda_1t}qquad
    mathbf{x}_2(t)=pmb{xi}^{(2)} e^{lambda_2t}qquad
    mathbf{x}_3(t)=pmb{xi}^{(3)} e^{lambda_3t}$$
    Then your fundamental matrix should be
    $$pmb{psi}(t)=left(
    begin{array}{@{}ccc@{}}
    mathbf{x}_1(t)&
    mathbf{x}_2(t)&
    mathbf{x}_3(t)
    end{array}right)=left(
    begin{array}{@{}ccc@{}}
    xi_1^{(1)e^{lambda_1t}}&xi_1^{(2)e^{lambda_2t}}&xi_1^{(3)e^{lambda_3t}}\
    xi_2^{(1)e^{lambda_1t}}&xi_2^{(2)e^{lambda_2t}}&xi_2^{(3)e^{lambda_3t}}\
    xi_3^{(1)e^{lambda_1t}}&xi_3^{(2)e^{lambda_2t}}&xi_3^{(3)e^{lambda_3t}}\
    end{array}
    right)
    $$
    where $xi_n^{(m)}$ denotes the $n$th element of $pmb{xi}^{(m)}$. Note that the general solution of the differential equation is
    $$mathbf{x}=pmb{psi}(t)mathbf{c}$$
    where $mathbf{c}=(c_1,c_2,c_3)^intercal$ is a constant matrix.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Assuming you have three unique eigenvalues $(lambda_1,lambda_2,lambda_3)$ and eigenvectors $(pmb{xi}^{(1)},pmb{xi}^{(2)},pmb{xi}^{(3)})$ you should have three linearly independent solutions in the form
      $$mathbf{x}_1(t)=pmb{xi}^{(1)} e^{lambda_1t}qquad
      mathbf{x}_2(t)=pmb{xi}^{(2)} e^{lambda_2t}qquad
      mathbf{x}_3(t)=pmb{xi}^{(3)} e^{lambda_3t}$$
      Then your fundamental matrix should be
      $$pmb{psi}(t)=left(
      begin{array}{@{}ccc@{}}
      mathbf{x}_1(t)&
      mathbf{x}_2(t)&
      mathbf{x}_3(t)
      end{array}right)=left(
      begin{array}{@{}ccc@{}}
      xi_1^{(1)e^{lambda_1t}}&xi_1^{(2)e^{lambda_2t}}&xi_1^{(3)e^{lambda_3t}}\
      xi_2^{(1)e^{lambda_1t}}&xi_2^{(2)e^{lambda_2t}}&xi_2^{(3)e^{lambda_3t}}\
      xi_3^{(1)e^{lambda_1t}}&xi_3^{(2)e^{lambda_2t}}&xi_3^{(3)e^{lambda_3t}}\
      end{array}
      right)
      $$
      where $xi_n^{(m)}$ denotes the $n$th element of $pmb{xi}^{(m)}$. Note that the general solution of the differential equation is
      $$mathbf{x}=pmb{psi}(t)mathbf{c}$$
      where $mathbf{c}=(c_1,c_2,c_3)^intercal$ is a constant matrix.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Assuming you have three unique eigenvalues $(lambda_1,lambda_2,lambda_3)$ and eigenvectors $(pmb{xi}^{(1)},pmb{xi}^{(2)},pmb{xi}^{(3)})$ you should have three linearly independent solutions in the form
        $$mathbf{x}_1(t)=pmb{xi}^{(1)} e^{lambda_1t}qquad
        mathbf{x}_2(t)=pmb{xi}^{(2)} e^{lambda_2t}qquad
        mathbf{x}_3(t)=pmb{xi}^{(3)} e^{lambda_3t}$$
        Then your fundamental matrix should be
        $$pmb{psi}(t)=left(
        begin{array}{@{}ccc@{}}
        mathbf{x}_1(t)&
        mathbf{x}_2(t)&
        mathbf{x}_3(t)
        end{array}right)=left(
        begin{array}{@{}ccc@{}}
        xi_1^{(1)e^{lambda_1t}}&xi_1^{(2)e^{lambda_2t}}&xi_1^{(3)e^{lambda_3t}}\
        xi_2^{(1)e^{lambda_1t}}&xi_2^{(2)e^{lambda_2t}}&xi_2^{(3)e^{lambda_3t}}\
        xi_3^{(1)e^{lambda_1t}}&xi_3^{(2)e^{lambda_2t}}&xi_3^{(3)e^{lambda_3t}}\
        end{array}
        right)
        $$
        where $xi_n^{(m)}$ denotes the $n$th element of $pmb{xi}^{(m)}$. Note that the general solution of the differential equation is
        $$mathbf{x}=pmb{psi}(t)mathbf{c}$$
        where $mathbf{c}=(c_1,c_2,c_3)^intercal$ is a constant matrix.






        share|cite|improve this answer









        $endgroup$



        Assuming you have three unique eigenvalues $(lambda_1,lambda_2,lambda_3)$ and eigenvectors $(pmb{xi}^{(1)},pmb{xi}^{(2)},pmb{xi}^{(3)})$ you should have three linearly independent solutions in the form
        $$mathbf{x}_1(t)=pmb{xi}^{(1)} e^{lambda_1t}qquad
        mathbf{x}_2(t)=pmb{xi}^{(2)} e^{lambda_2t}qquad
        mathbf{x}_3(t)=pmb{xi}^{(3)} e^{lambda_3t}$$
        Then your fundamental matrix should be
        $$pmb{psi}(t)=left(
        begin{array}{@{}ccc@{}}
        mathbf{x}_1(t)&
        mathbf{x}_2(t)&
        mathbf{x}_3(t)
        end{array}right)=left(
        begin{array}{@{}ccc@{}}
        xi_1^{(1)e^{lambda_1t}}&xi_1^{(2)e^{lambda_2t}}&xi_1^{(3)e^{lambda_3t}}\
        xi_2^{(1)e^{lambda_1t}}&xi_2^{(2)e^{lambda_2t}}&xi_2^{(3)e^{lambda_3t}}\
        xi_3^{(1)e^{lambda_1t}}&xi_3^{(2)e^{lambda_2t}}&xi_3^{(3)e^{lambda_3t}}\
        end{array}
        right)
        $$
        where $xi_n^{(m)}$ denotes the $n$th element of $pmb{xi}^{(m)}$. Note that the general solution of the differential equation is
        $$mathbf{x}=pmb{psi}(t)mathbf{c}$$
        where $mathbf{c}=(c_1,c_2,c_3)^intercal$ is a constant matrix.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 2 '16 at 16:45









        cheesyfluffcheesyfluff

        1,482520




        1,482520






























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