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I am computing the exponential of a matrix via Taylor expansion to prove the end-result with induction.

For a matrix $A=PDP^{-1}$ where $D$ is the diagonalized matrix, is there any kind of formula for adding their sums like in the instance of

$$I+PDP^{-1}+\frac{PD^{2}P^{-1}}{2!}+\frac{PD^{3}P^{-1}}{3!}...?$$

Alekos Robotis
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Vane Voe
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    Factor it out as $P\left(I + D + \frac{D^2}{2!} + \frac{D^3}{3!} + \ldots\right)P^{-1}$. The stuff between parentheses consists of sums over the diagonal entries which are just the eigenvalues $\lambda_i$ of $A$, and the entry-wise sum converges to a diagonal matrix with entries equal to $e^{\lambda_i}$. Of course you would first need to define a metric on the space of matrices, and show that limits commute with multiplication. – Tob Ernack Feb 08 '19 at 06:18
  • Of course, good observation. If you made that into an answer I would accept it. – Vane Voe Feb 08 '19 at 07:49
  • Related question. – Servaes Feb 08 '19 at 09:47

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