TITLE: Matrix Exponential of Identity Matrix
QUESTION [5 upvotes]: I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? How should I calculate $e^{I_n}$

REPLY [9 votes]: You can use the Taylor expansion of the exponential,
$\displaystyle e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$.
All the powers of the identity matrix are equal to the identity matrix, so $\displaystyle I_n^k = I_n$.
Therefore,
$\displaystyle e^{I_n} = I_n \sum_{k=0}^\infty \frac{1}{k!} = e \, I_n$.
This is a diagonal matrix with all the terms in the diagonal being $e$.