How to transform an expression into a form involving the trace of a product of two matrices












3












$begingroup$


In page 594 of Bishop's PRML, the following equation is implied:



$$
-frac{1}{2}sum(mathbf{x}_n-mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n-mathbf{bar{x}}) = -frac{N}{2}mathrm{Tr}(mathbf{C}^{-1}mathbf{S})
$$



where



$$
mathbf{S} = frac{1}{N}sum(mathbf{x}_n-mathbf{bar{x}})(mathbf{x}_n-mathbf{bar{x}})^T
$$
,$mathbf{C}$ is a symmetric matrix and $mathbf{bar{x}} = frac{sum_{n=1}^Nmathbf{x}_n}{N}$.



I want to derive this equation myself. But I'm not sure how to do it. Could someone show why the equation holds?






linear-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    I have taken the liberty to modify your title which was "uninformative".
    $endgroup$
    – Jean Marie
    Dec 31 '18 at 19:48
















3












$begingroup$


In page 594 of Bishop's PRML, the following equation is implied:



$$
-frac{1}{2}sum(mathbf{x}_n-mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n-mathbf{bar{x}}) = -frac{N}{2}mathrm{Tr}(mathbf{C}^{-1}mathbf{S})
$$



where



$$
mathbf{S} = frac{1}{N}sum(mathbf{x}_n-mathbf{bar{x}})(mathbf{x}_n-mathbf{bar{x}})^T
$$
,$mathbf{C}$ is a symmetric matrix and $mathbf{bar{x}} = frac{sum_{n=1}^Nmathbf{x}_n}{N}$.



I want to derive this equation myself. But I'm not sure how to do it. Could someone show why the equation holds?






linear-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    I have taken the liberty to modify your title which was "uninformative".
    $endgroup$
    – Jean Marie
    Dec 31 '18 at 19:48














3












3








3





$begingroup$


In page 594 of Bishop's PRML, the following equation is implied:



$$
-frac{1}{2}sum(mathbf{x}_n-mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n-mathbf{bar{x}}) = -frac{N}{2}mathrm{Tr}(mathbf{C}^{-1}mathbf{S})
$$



where



$$
mathbf{S} = frac{1}{N}sum(mathbf{x}_n-mathbf{bar{x}})(mathbf{x}_n-mathbf{bar{x}})^T
$$
,$mathbf{C}$ is a symmetric matrix and $mathbf{bar{x}} = frac{sum_{n=1}^Nmathbf{x}_n}{N}$.



I want to derive this equation myself. But I'm not sure how to do it. Could someone show why the equation holds?






linear-algebra







share|cite|improve this question











$endgroup$




In page 594 of Bishop's PRML, the following equation is implied:



$$
-frac{1}{2}sum(mathbf{x}_n-mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n-mathbf{bar{x}}) = -frac{N}{2}mathrm{Tr}(mathbf{C}^{-1}mathbf{S})
$$



where



$$
mathbf{S} = frac{1}{N}sum(mathbf{x}_n-mathbf{bar{x}})(mathbf{x}_n-mathbf{bar{x}})^T
$$
,$mathbf{C}$ is a symmetric matrix and $mathbf{bar{x}} = frac{sum_{n=1}^Nmathbf{x}_n}{N}$.



I want to derive this equation myself. But I'm not sure how to do it. Could someone show why the equation holds?






linear-algebra




linear-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 31 '18 at 19:46









Jean Marie

31.6k42355




31.6k42355










asked Dec 31 '18 at 12:16









SandiSandi

262112




262112












  • $begingroup$
    I have taken the liberty to modify your title which was "uninformative".
    $endgroup$
    – Jean Marie
    Dec 31 '18 at 19:48


















  • $begingroup$
    I have taken the liberty to modify your title which was "uninformative".
    $endgroup$
    – Jean Marie
    Dec 31 '18 at 19:48
















$begingroup$
I have taken the liberty to modify your title which was "uninformative".
$endgroup$
– Jean Marie
Dec 31 '18 at 19:48




$begingroup$
I have taken the liberty to modify your title which was "uninformative".
$endgroup$
– Jean Marie
Dec 31 '18 at 19:48










2 Answers
2






active

oldest

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4












$begingroup$

Guide:



Notice that $(x_n - bar{x})^TC^{-1}(x_n - bar{x})$ is a scalar,



hence $$(x_n - bar{x})^TC^{-1}(x_n - bar{x})= operatorname{Tr}left[(x_n - bar{x})^TC^{-1}(x_n - bar{x})right]=operatorname{Tr}left[C^{-1}(x_n - bar{x})(x_n - bar{x})^Tright]$$



since $operatorname{Tr}(AB)=operatorname{Tr}(BA)$.
Hopefully you can take it from here.






share|cite|improve this answer











$endgroup$





















    4












    $begingroup$

    With the help of Siong Thye Goh, I did the following:



    begin{align}
    -frac{1}{2}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}}) &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})]\
    &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
    &= -frac{1}{2}mathrm{Tr}[sum^N_{n=1}mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
    &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
    &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}Nmathbf{S}] = -frac{N}{2}mathrm{Tr}[mathbf{C}^{-1}mathbf{S}]
    end{align}






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
      $endgroup$
      – Siong Thye Goh
      Dec 31 '18 at 13:00












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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    Guide:



    Notice that $(x_n - bar{x})^TC^{-1}(x_n - bar{x})$ is a scalar,



    hence $$(x_n - bar{x})^TC^{-1}(x_n - bar{x})= operatorname{Tr}left[(x_n - bar{x})^TC^{-1}(x_n - bar{x})right]=operatorname{Tr}left[C^{-1}(x_n - bar{x})(x_n - bar{x})^Tright]$$



    since $operatorname{Tr}(AB)=operatorname{Tr}(BA)$.
    Hopefully you can take it from here.






    share|cite|improve this answer











    $endgroup$


















      4












      $begingroup$

      Guide:



      Notice that $(x_n - bar{x})^TC^{-1}(x_n - bar{x})$ is a scalar,



      hence $$(x_n - bar{x})^TC^{-1}(x_n - bar{x})= operatorname{Tr}left[(x_n - bar{x})^TC^{-1}(x_n - bar{x})right]=operatorname{Tr}left[C^{-1}(x_n - bar{x})(x_n - bar{x})^Tright]$$



      since $operatorname{Tr}(AB)=operatorname{Tr}(BA)$.
      Hopefully you can take it from here.






      share|cite|improve this answer











      $endgroup$
















        4












        4








        4





        $begingroup$

        Guide:



        Notice that $(x_n - bar{x})^TC^{-1}(x_n - bar{x})$ is a scalar,



        hence $$(x_n - bar{x})^TC^{-1}(x_n - bar{x})= operatorname{Tr}left[(x_n - bar{x})^TC^{-1}(x_n - bar{x})right]=operatorname{Tr}left[C^{-1}(x_n - bar{x})(x_n - bar{x})^Tright]$$



        since $operatorname{Tr}(AB)=operatorname{Tr}(BA)$.
        Hopefully you can take it from here.






        share|cite|improve this answer











        $endgroup$



        Guide:



        Notice that $(x_n - bar{x})^TC^{-1}(x_n - bar{x})$ is a scalar,



        hence $$(x_n - bar{x})^TC^{-1}(x_n - bar{x})= operatorname{Tr}left[(x_n - bar{x})^TC^{-1}(x_n - bar{x})right]=operatorname{Tr}left[C^{-1}(x_n - bar{x})(x_n - bar{x})^Tright]$$



        since $operatorname{Tr}(AB)=operatorname{Tr}(BA)$.
        Hopefully you can take it from here.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 31 '18 at 12:28









        Bernard

        125k743119




        125k743119










        answered Dec 31 '18 at 12:19









        Siong Thye GohSiong Thye Goh

        105k1469121




        105k1469121























            4












            $begingroup$

            With the help of Siong Thye Goh, I did the following:



            begin{align}
            -frac{1}{2}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}}) &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})]\
            &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[sum^N_{n=1}mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}Nmathbf{S}] = -frac{N}{2}mathrm{Tr}[mathbf{C}^{-1}mathbf{S}]
            end{align}






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
              $endgroup$
              – Siong Thye Goh
              Dec 31 '18 at 13:00
















            4












            $begingroup$

            With the help of Siong Thye Goh, I did the following:



            begin{align}
            -frac{1}{2}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}}) &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})]\
            &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[sum^N_{n=1}mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}Nmathbf{S}] = -frac{N}{2}mathrm{Tr}[mathbf{C}^{-1}mathbf{S}]
            end{align}






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
              $endgroup$
              – Siong Thye Goh
              Dec 31 '18 at 13:00














            4












            4








            4





            $begingroup$

            With the help of Siong Thye Goh, I did the following:



            begin{align}
            -frac{1}{2}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}}) &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})]\
            &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[sum^N_{n=1}mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}Nmathbf{S}] = -frac{N}{2}mathrm{Tr}[mathbf{C}^{-1}mathbf{S}]
            end{align}






            share|cite|improve this answer











            $endgroup$



            With the help of Siong Thye Goh, I did the following:



            begin{align}
            -frac{1}{2}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}}) &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[(mathbf{x}_n - mathbf{bar{x}})^Tmathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})]\
            &= -frac{1}{2}sum_{n=1}^Nmathrm{Tr}[mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[sum^N_{n=1}mathbf{C}^{-1}(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}sum_{n=1}^N(mathbf{x}_n - mathbf{bar{x}})(mathbf{x}_n - mathbf{bar{x}})^T]\
            &= -frac{1}{2}mathrm{Tr}[mathbf{C}^{-1}Nmathbf{S}] = -frac{N}{2}mathrm{Tr}[mathbf{C}^{-1}mathbf{S}]
            end{align}







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 1 at 14:34

























            answered Dec 31 '18 at 12:43









            SandiSandi

            262112




            262112












            • $begingroup$
              +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
              $endgroup$
              – Siong Thye Goh
              Dec 31 '18 at 13:00


















            • $begingroup$
              +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
              $endgroup$
              – Siong Thye Goh
              Dec 31 '18 at 13:00
















            $begingroup$
            +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
            $endgroup$
            – Siong Thye Goh
            Dec 31 '18 at 13:00




            $begingroup$
            +1 great job! just a minor careless step at the very beginning, there's a negative there. ;)
            $endgroup$
            – Siong Thye Goh
            Dec 31 '18 at 13:00


















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