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I have a random $3\times 3$ matrix $A$.

How can I calculate $e^A$ by $E$ (the identity matrix), $A$ and $A^2$, using the Cayley-Hamilton theorem?

I need a general expression that includes only the trace, determinant and signature (tensor invariants) of a given matrix $A$.

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    $\begingroup$ Is there a specific reason that you want to use the Cayley Hamilton theorem, or are you merely trying to express that you want an expression of the form $e^A = c_1 E + c_2 A + c_3 A^2$? $\endgroup$ Commented Mar 10, 2023 at 20:06
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    $\begingroup$ Also, please explicitly state what you mean by $E$. I strongly suspect that you mean the identity matrix. $\endgroup$ Commented Mar 10, 2023 at 20:07
  • $\begingroup$ Finally, note that askers are expected to provide context for their questions, as is explained here. Please edit your question to tell us why you are asking this question (e.g. is this an approach to a research problem? A homework problem of some kind?), what you have tried so far, and any other relevant thoughts you have. $\endgroup$ Commented Mar 10, 2023 at 20:09
  • $\begingroup$ I really want an expression of the form $e^A = c_1 E + c_2 A + c_3 A^2$ . I know for sure that such a decomposition is possible precisely with the help of the Cayley-Hamilton theorem. $\endgroup$ Commented Mar 10, 2023 at 23:00

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