derivative of an integral with respect to matrix element
$begingroup$
I need to compute the following derivative:
$$
frac{partial}{partial V_{mn}} int prod_j dphi_j , mathrm{Tr}
left(ln(i phi + beta J V)right)
$$
where $phi equiv phi_i delta_{ij}$ is a diagonal matrix and $V$ is a matrix with elements $V_{mn}$ that are independent of $phi_j$ (note the $i$ inside the natural log is the imaginary unit).
My attempt
1) I understand the $V_{mn}$ and $phi_j$ are independent, so the derivative can be carried into the integral.
2) I'm less sure about whether I can carry the derivative into the trace -- but I assume so because of linearity.
Then I would think the answer would be (with chain rule):
$$
int prod_j dphi_j , beta J (iphi + beta J V)^{-1}_{mn}
$$
Is my reasoning/answer correct? Any hints appreciated.
linear-algebra partial-derivative matrix-calculus trace
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
$endgroup$
add a comment |
$begingroup$
I need to compute the following derivative:
$$
frac{partial}{partial V_{mn}} int prod_j dphi_j , mathrm{Tr}
left(ln(i phi + beta J V)right)
$$
where $phi equiv phi_i delta_{ij}$ is a diagonal matrix and $V$ is a matrix with elements $V_{mn}$ that are independent of $phi_j$ (note the $i$ inside the natural log is the imaginary unit).
My attempt
1) I understand the $V_{mn}$ and $phi_j$ are independent, so the derivative can be carried into the integral.
2) I'm less sure about whether I can carry the derivative into the trace -- but I assume so because of linearity.
Then I would think the answer would be (with chain rule):
$$
int prod_j dphi_j , beta J (iphi + beta J V)^{-1}_{mn}
$$
Is my reasoning/answer correct? Any hints appreciated.
linear-algebra partial-derivative matrix-calculus trace
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
$endgroup$
$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57
add a comment |
$begingroup$
I need to compute the following derivative:
$$
frac{partial}{partial V_{mn}} int prod_j dphi_j , mathrm{Tr}
left(ln(i phi + beta J V)right)
$$
where $phi equiv phi_i delta_{ij}$ is a diagonal matrix and $V$ is a matrix with elements $V_{mn}$ that are independent of $phi_j$ (note the $i$ inside the natural log is the imaginary unit).
My attempt
1) I understand the $V_{mn}$ and $phi_j$ are independent, so the derivative can be carried into the integral.
2) I'm less sure about whether I can carry the derivative into the trace -- but I assume so because of linearity.
Then I would think the answer would be (with chain rule):
$$
int prod_j dphi_j , beta J (iphi + beta J V)^{-1}_{mn}
$$
Is my reasoning/answer correct? Any hints appreciated.
linear-algebra partial-derivative matrix-calculus trace
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
$endgroup$
I need to compute the following derivative:
$$
frac{partial}{partial V_{mn}} int prod_j dphi_j , mathrm{Tr}
left(ln(i phi + beta J V)right)
$$
where $phi equiv phi_i delta_{ij}$ is a diagonal matrix and $V$ is a matrix with elements $V_{mn}$ that are independent of $phi_j$ (note the $i$ inside the natural log is the imaginary unit).
My attempt
1) I understand the $V_{mn}$ and $phi_j$ are independent, so the derivative can be carried into the integral.
2) I'm less sure about whether I can carry the derivative into the trace -- but I assume so because of linearity.
Then I would think the answer would be (with chain rule):
$$
int prod_j dphi_j , beta J (iphi + beta J V)^{-1}_{mn}
$$
Is my reasoning/answer correct? Any hints appreciated.
linear-algebra partial-derivative matrix-calculus trace
linear-algebra partial-derivative matrix-calculus trace
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
edited Dec 19 '18 at 4:27
ksgj1
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
asked Dec 18 '18 at 21:23
ksgj1ksgj1
1356
1356
$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57
add a comment |
$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57
$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57
add a comment |
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$begingroup$
Replace the expression $,beta J(iphi+beta JV)^{-1},$ by $,,beta J^T(iphi+beta JV)^{-T}.$
$endgroup$
– greg
Dec 18 '18 at 22:26
$begingroup$
@greg 1) What does $-T$ as an exponent mean? inverse + transpose? 2) $J$ is a number, not a matrix. 3) why and how does the transpose come into play?
$endgroup$
– ksgj1
Dec 18 '18 at 22:57