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I need to take the derivative of an exponential function with respect to a matrix.

$\frac{\partial}{\partial A} (e^{At})$

where $t$ is a scalar and $A$ is a matrix not dependent on $t$. Yes those $A$'s are the same variable for clarification.

Thank you

c smith
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  • What is your definition for matrix derivative? – Batman Jul 12 '16 at 18:48
  • You should be able to carry out the derivative as you would do for the scalar case...For that one should use the definition of the matrix exponential – Alex Jul 12 '16 at 18:55
  • I'm not sure how best to define that. how about The derivative with respect to the elements of the matrix $A$. – c smith Jul 12 '16 at 18:55
  • Ok. Here is a better question. If I perform a series expansion of the the matrix exponential and take its derivative with respect to $A$ is the answer $\sum_{k=1}^{\inf}A^{k-1}\frac{t^k}{k!}$ – c smith Jul 12 '16 at 18:57
  • You can't take the derivitive using the limit definition, as not all nonzero matrices are invertible. So I suggest taking the infinite series, taking the derivitive treating as scalars, and then making it back a function of matrices. So we gain, informally $\frac{\delta}{\delta A} e^{At}=t e^{At}$. – Mar Jul 12 '16 at 19:01
  • Be aware that $\partial_A(A^2)(h)=hA+Ah$ – Tsemo Aristide Jul 12 '16 at 19:03
  • Take a look at this question: http://math.stackexchange.com/questions/1291063/derivative-of-matrix-exponential-wrt-to-each-element-of-matrix – frank Jul 13 '16 at 00:32

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