Finding the hessian matrix of $f(x) = g^t(x)g(x)$

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Let $f:mathbbR^n to mathbbR$ defined by $f(x)=g^t(x)g(x)$, where $g:mathbbR^n to mathbbR^m$ and $x in mathbbR^n$.
Find the Hessian of $f$.



(The $^t$ means transpose)



My attempt:



The k-th element of gradient vector is:



$displaystyle f(x) = g^t(x)g(x) = sum_i=1^m g_i^2(x)$



$displaystyle fracpartialpartial x_k sum_i=1^m g_i^2(x)= sum_i=1^m fracpartialpartial x_kg_i^2(x)=sum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x)$



Then $nabla f(x) = 2g^t(x)Jg(x)$



The mk-th element of hessian matrix is:



$displaystyle dfracpartialpartial x_msum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m dfracpartialpartial x_mg_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m bigg(dfracpartialpartial x_mg_i(x)bigg)fracpartialpartial x_kg_i(x) +g_i(x)bigg(dfracpartial^2partial x_mpartial x_kg_i(x)bigg)$



Is this correct?
What matrix could then be this hessian?



Thanks




multivariable-calculus vector-analysis




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  • is it $2(J^2g(x) + g^t(x)Hg(x))$?
    – dude3221
    Aug 18 at 3:10














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0
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Let $f:mathbbR^n to mathbbR$ defined by $f(x)=g^t(x)g(x)$, where $g:mathbbR^n to mathbbR^m$ and $x in mathbbR^n$.
Find the Hessian of $f$.



(The $^t$ means transpose)



My attempt:



The k-th element of gradient vector is:



$displaystyle f(x) = g^t(x)g(x) = sum_i=1^m g_i^2(x)$



$displaystyle fracpartialpartial x_k sum_i=1^m g_i^2(x)= sum_i=1^m fracpartialpartial x_kg_i^2(x)=sum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x)$



Then $nabla f(x) = 2g^t(x)Jg(x)$



The mk-th element of hessian matrix is:



$displaystyle dfracpartialpartial x_msum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m dfracpartialpartial x_mg_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m bigg(dfracpartialpartial x_mg_i(x)bigg)fracpartialpartial x_kg_i(x) +g_i(x)bigg(dfracpartial^2partial x_mpartial x_kg_i(x)bigg)$



Is this correct?
What matrix could then be this hessian?



Thanks




multivariable-calculus vector-analysis




share|cite|improve this question




















  • is it $2(J^2g(x) + g^t(x)Hg(x))$?
    – dude3221
    Aug 18 at 3:10












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $f:mathbbR^n to mathbbR$ defined by $f(x)=g^t(x)g(x)$, where $g:mathbbR^n to mathbbR^m$ and $x in mathbbR^n$.
Find the Hessian of $f$.



(The $^t$ means transpose)



My attempt:



The k-th element of gradient vector is:



$displaystyle f(x) = g^t(x)g(x) = sum_i=1^m g_i^2(x)$



$displaystyle fracpartialpartial x_k sum_i=1^m g_i^2(x)= sum_i=1^m fracpartialpartial x_kg_i^2(x)=sum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x)$



Then $nabla f(x) = 2g^t(x)Jg(x)$



The mk-th element of hessian matrix is:



$displaystyle dfracpartialpartial x_msum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m dfracpartialpartial x_mg_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m bigg(dfracpartialpartial x_mg_i(x)bigg)fracpartialpartial x_kg_i(x) +g_i(x)bigg(dfracpartial^2partial x_mpartial x_kg_i(x)bigg)$



Is this correct?
What matrix could then be this hessian?



Thanks




multivariable-calculus vector-analysis




share|cite|improve this question












Let $f:mathbbR^n to mathbbR$ defined by $f(x)=g^t(x)g(x)$, where $g:mathbbR^n to mathbbR^m$ and $x in mathbbR^n$.
Find the Hessian of $f$.



(The $^t$ means transpose)



My attempt:



The k-th element of gradient vector is:



$displaystyle f(x) = g^t(x)g(x) = sum_i=1^m g_i^2(x)$



$displaystyle fracpartialpartial x_k sum_i=1^m g_i^2(x)= sum_i=1^m fracpartialpartial x_kg_i^2(x)=sum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x)$



Then $nabla f(x) = 2g^t(x)Jg(x)$



The mk-th element of hessian matrix is:



$displaystyle dfracpartialpartial x_msum_i=1^m2 g_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m dfracpartialpartial x_mg_i(x)fracpartialpartial x_kg_i(x) = 2sum_i=1^m bigg(dfracpartialpartial x_mg_i(x)bigg)fracpartialpartial x_kg_i(x) +g_i(x)bigg(dfracpartial^2partial x_mpartial x_kg_i(x)bigg)$



Is this correct?
What matrix could then be this hessian?



Thanks





multivariable-calculus vector-analysis





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 18 at 2:45









dude3221

31812




31812











  • is it $2(J^2g(x) + g^t(x)Hg(x))$?
    – dude3221
    Aug 18 at 3:10
















  • is it $2(J^2g(x) + g^t(x)Hg(x))$?
    – dude3221
    Aug 18 at 3:10















is it $2(J^2g(x) + g^t(x)Hg(x))$?
– dude3221
Aug 18 at 3:10




is it $2(J^2g(x) + g^t(x)Hg(x))$?
– dude3221
Aug 18 at 3:10















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