The Structure of the Jacobian Matrix in One-to-One Transformations












2












$begingroup$


I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $Theta rightarrowDelta$ as



$J_{Theta rightarrowDelta}=left[_m frac {partial Theta_i}{partial Delta_j}right]_{i,j}$



$m$ indicates that it is a matrix with the corresponding partial derivatives at $(i,j)$. Hence, $J$ is the Jacobian matrix of a function which takes as input the vector $Theta$ and produces as output the vector $Delta$. Or as the authors put it: parameters in $Theta$ "are transformed in a one-to-one manner to the vector $Delta$"



However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:



$ J=[frac {partial f}{partial x_i}...frac {partial f}{partial x_n}]$



Here, J is the Jacobian matrix $[m,n]$ of $f$, where $f:ℝ^n → ℝ^m$ is a function which takes as input the vector $x$ and produces as output the vector $f(x)$.



Hence, I find the two definitions of the Jacobian matrix conflicting since



$J_{Theta rightarrowDelta} = J^{-1}$



I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?






linear-algebra matrices jacobian







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $Theta rightarrowDelta$ as



    $J_{Theta rightarrowDelta}=left[_m frac {partial Theta_i}{partial Delta_j}right]_{i,j}$



    $m$ indicates that it is a matrix with the corresponding partial derivatives at $(i,j)$. Hence, $J$ is the Jacobian matrix of a function which takes as input the vector $Theta$ and produces as output the vector $Delta$. Or as the authors put it: parameters in $Theta$ "are transformed in a one-to-one manner to the vector $Delta$"



    However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:



    $ J=[frac {partial f}{partial x_i}...frac {partial f}{partial x_n}]$



    Here, J is the Jacobian matrix $[m,n]$ of $f$, where $f:ℝ^n → ℝ^m$ is a function which takes as input the vector $x$ and produces as output the vector $f(x)$.



    Hence, I find the two definitions of the Jacobian matrix conflicting since



    $J_{Theta rightarrowDelta} = J^{-1}$



    I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?






    linear-algebra matrices jacobian







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $Theta rightarrowDelta$ as



      $J_{Theta rightarrowDelta}=left[_m frac {partial Theta_i}{partial Delta_j}right]_{i,j}$



      $m$ indicates that it is a matrix with the corresponding partial derivatives at $(i,j)$. Hence, $J$ is the Jacobian matrix of a function which takes as input the vector $Theta$ and produces as output the vector $Delta$. Or as the authors put it: parameters in $Theta$ "are transformed in a one-to-one manner to the vector $Delta$"



      However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:



      $ J=[frac {partial f}{partial x_i}...frac {partial f}{partial x_n}]$



      Here, J is the Jacobian matrix $[m,n]$ of $f$, where $f:ℝ^n → ℝ^m$ is a function which takes as input the vector $x$ and produces as output the vector $f(x)$.



      Hence, I find the two definitions of the Jacobian matrix conflicting since



      $J_{Theta rightarrowDelta} = J^{-1}$



      I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?






      linear-algebra matrices jacobian







      share|cite|improve this question









      $endgroup$




      I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $Theta rightarrowDelta$ as



      $J_{Theta rightarrowDelta}=left[_m frac {partial Theta_i}{partial Delta_j}right]_{i,j}$



      $m$ indicates that it is a matrix with the corresponding partial derivatives at $(i,j)$. Hence, $J$ is the Jacobian matrix of a function which takes as input the vector $Theta$ and produces as output the vector $Delta$. Or as the authors put it: parameters in $Theta$ "are transformed in a one-to-one manner to the vector $Delta$"



      However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:



      $ J=[frac {partial f}{partial x_i}...frac {partial f}{partial x_n}]$



      Here, J is the Jacobian matrix $[m,n]$ of $f$, where $f:ℝ^n → ℝ^m$ is a function which takes as input the vector $x$ and produces as output the vector $f(x)$.



      Hence, I find the two definitions of the Jacobian matrix conflicting since



      $J_{Theta rightarrowDelta} = J^{-1}$



      I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?






      linear-algebra matrices jacobian




      linear-algebra matrices jacobian






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 24 at 20:13









      DomBDomB

      396




      396






















          1 Answer
          1






          active

          oldest

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          2












          $begingroup$

          That's just because of how the Jacobian is used in probability. If $Delta = f(Theta)$ and $p_Delta, p_Theta$ are the two densities then



          $$mathbf{P}(Delta in A) = int_A p_{Delta}(delta) ,ddelta = mathbf{P}(Theta in f^{-1}(A)) = int_{f^{-1}(A)} p_Theta(theta) ,dtheta $$



          The change of variables theorem tells us that



          $$ int_{f^{-1}(A)} p_Theta(theta) ,dtheta = int_A p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right| ,ddelta. $$



          So



          $$ p_Delta(delta) = p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right|. $$



          Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
            $endgroup$
            – DomB
            Jan 25 at 18:47










          • $begingroup$
            @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
            $endgroup$
            – Trevor Gunn
            Jan 25 at 22:11











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          That's just because of how the Jacobian is used in probability. If $Delta = f(Theta)$ and $p_Delta, p_Theta$ are the two densities then



          $$mathbf{P}(Delta in A) = int_A p_{Delta}(delta) ,ddelta = mathbf{P}(Theta in f^{-1}(A)) = int_{f^{-1}(A)} p_Theta(theta) ,dtheta $$



          The change of variables theorem tells us that



          $$ int_{f^{-1}(A)} p_Theta(theta) ,dtheta = int_A p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right| ,ddelta. $$



          So



          $$ p_Delta(delta) = p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right|. $$



          Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
            $endgroup$
            – DomB
            Jan 25 at 18:47










          • $begingroup$
            @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
            $endgroup$
            – Trevor Gunn
            Jan 25 at 22:11
















          2












          $begingroup$

          That's just because of how the Jacobian is used in probability. If $Delta = f(Theta)$ and $p_Delta, p_Theta$ are the two densities then



          $$mathbf{P}(Delta in A) = int_A p_{Delta}(delta) ,ddelta = mathbf{P}(Theta in f^{-1}(A)) = int_{f^{-1}(A)} p_Theta(theta) ,dtheta $$



          The change of variables theorem tells us that



          $$ int_{f^{-1}(A)} p_Theta(theta) ,dtheta = int_A p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right| ,ddelta. $$



          So



          $$ p_Delta(delta) = p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right|. $$



          Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
            $endgroup$
            – DomB
            Jan 25 at 18:47










          • $begingroup$
            @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
            $endgroup$
            – Trevor Gunn
            Jan 25 at 22:11














          2












          2








          2





          $begingroup$

          That's just because of how the Jacobian is used in probability. If $Delta = f(Theta)$ and $p_Delta, p_Theta$ are the two densities then



          $$mathbf{P}(Delta in A) = int_A p_{Delta}(delta) ,ddelta = mathbf{P}(Theta in f^{-1}(A)) = int_{f^{-1}(A)} p_Theta(theta) ,dtheta $$



          The change of variables theorem tells us that



          $$ int_{f^{-1}(A)} p_Theta(theta) ,dtheta = int_A p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right| ,ddelta. $$



          So



          $$ p_Delta(delta) = p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right|. $$



          Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.






          share|cite|improve this answer









          $endgroup$



          That's just because of how the Jacobian is used in probability. If $Delta = f(Theta)$ and $p_Delta, p_Theta$ are the two densities then



          $$mathbf{P}(Delta in A) = int_A p_{Delta}(delta) ,ddelta = mathbf{P}(Theta in f^{-1}(A)) = int_{f^{-1}(A)} p_Theta(theta) ,dtheta $$



          The change of variables theorem tells us that



          $$ int_{f^{-1}(A)} p_Theta(theta) ,dtheta = int_A p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right| ,ddelta. $$



          So



          $$ p_Delta(delta) = p_Theta(f^{-1}(delta))left| detleft[ frac{partial Theta_i}{partial Delta_j} right] right|. $$



          Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 24 at 20:48









          Trevor GunnTrevor Gunn

          14.9k32047




          14.9k32047












          • $begingroup$
            thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
            $endgroup$
            – DomB
            Jan 25 at 18:47










          • $begingroup$
            @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
            $endgroup$
            – Trevor Gunn
            Jan 25 at 22:11


















          • $begingroup$
            thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
            $endgroup$
            – DomB
            Jan 25 at 18:47










          • $begingroup$
            @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
            $endgroup$
            – Trevor Gunn
            Jan 25 at 22:11
















          $begingroup$
          thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
          $endgroup$
          – DomB
          Jan 25 at 18:47




          $begingroup$
          thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers
          $endgroup$
          – DomB
          Jan 25 at 18:47












          $begingroup$
          @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
          $endgroup$
          – Trevor Gunn
          Jan 25 at 22:11




          $begingroup$
          @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds).
          $endgroup$
          – Trevor Gunn
          Jan 25 at 22:11


















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