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The matrix is

$$ A = \begin{pmatrix} 4 & -2& \\ 3 & -1&\\ \end{pmatrix} $$

And I need to caluclate $\exp(At)$, where $t$ is real.

Not sure what I need to do after I multiply everything by $t$

please help :)

E. Joseph
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john la
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  • Possible duplicate ? There have been extensive discussions on matrix functions. See e.g. http://math.stackexchange.com/questions/1634488/how-could-we-define-the-factorial-of-a-matrix/1980621#1980621, http://mathematica.stackexchange.com/questions/128408/why-does-matrixfunction-with-sinc-return-this-error/128437#128437. See also the "Related" section. – Dr. Wolfgang Hintze Nov 01 '16 at 10:44
  • Why $&$ before $\backslash\backslash$ in the matrix? – Did Nov 02 '16 at 15:47

2 Answers2

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Hint

  • Diagonalize $At$, which will give you $P$ such that $At=PDP^{-1}$.

  • Remark that if $\exp(At)=P\exp(D)P^{-1}$.

  • Since $D$ is diagonal, $\exp(D)$ is easy to calculate.

E. Joseph
  • 14,843
  • Actually, it is sufficient to diagonalize just $\mathbf A$, since for a scalar $t$, $t \mathbf A=t \mathbf P \mathbf D \mathbf {P^{-1}}=\mathbf P t \mathbf D \mathbf {P^{-1}}$ – G Cab Nov 01 '16 at 11:18
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You can use the Laplace transform; $$\exp(At)=\mathscr{L}^{-1}\{ (sI-A)^{-1} \}.$$

Mehran
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