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I need to find a way to solve the following problem:

Suppose I have a singular matrix A. I need to find a solution to

$\frac{e^{hA}-I}{A}$

However, note that the inverse of A is infinity. Can anyone help with this problem? Maybe l'hospital rule could help? I am not really sure. Thanks in advance.

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  • $\begingroup$ Welcome to MSE. If you say that you want to find a solution, you should write down an equation. Moreover, let us know what you have tried already and where your interest in this question comes from. $\endgroup$ – user526015 Dec 4 '18 at 14:12
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What you wrote doesn't make sense, however the function $f(z) = \dfrac{e^{hz}-1}{z}$ has a removable singularity at $0$, and after removing it you have an entire function, so you can define $f(A)$ by the functional calculus. For example, you could compute it using the power series $$ f(A) = \sum_{n=1}^\infty \frac{h^n}{n!} A^{n-1} $$ Or if $A$ is diagonalizable: $A = S^{-1} D S$ where $D$ is diagonal, then $f(A) = S^{-1} f(D) S$ where $f(D)$ is diagonal with $(f(D))_{jj} = f(D_{jj})$.

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  • $\begingroup$ Thank you very much for your response. Sorry to bother again, but just to be clear, there is no way to get rid of the power series? I have used software to compute this series and it is a convergent series, therefore I would like to find a simple way to compute it which would not involve a summation of infinity terms! $\endgroup$ – Thiago Lima Dec 4 '18 at 14:59
  • $\begingroup$ There are many ways to compute it besides the series. I noted one in the case where $A$ is diagonalizable. More generally you could use Jordan canonical form. $\endgroup$ – Robert Israel Dec 4 '18 at 16:53
  • $\begingroup$ See also Moler and van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later $\endgroup$ – Robert Israel Dec 4 '18 at 17:07
  • $\begingroup$ Ok, I will look into it. Thank you very much again! $\endgroup$ – Thiago Lima Dec 4 '18 at 18:56

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