Can we define a exponential operation between scalars and matrices?

Question

I was wondering about a few operations we can perform between scalar and matrices (or vectors). We have a well defined multiplication, subtraction and addition. But i was wondering: if $A$ is a $n \times n$ matrix, can we define something like $e^A$? How can we calculate that? My concept of exponential is that $e^n = e \times e \times e...$ $n$ times. But what would it mean $A$ times? I did a little bit of searching and found some questions about matrix exponentials, but no definition. Thanks.

Answer 1

The matrix exponential of an $n\times n$ matrix $A$ is defined as: $$e^A=\sum_{n=0}^\infty\frac{A^n}{n!}$$ which uses the power series for $e^x$ where $x=A$. You can check out this definition and more here. Note that $A^0=I_n$.

Answer 2

We can indeed define the exponential of a matrix. We can generalize this further to the exponential of linear operators (bounded or unbounded even!). We can start by defining the exponential of a diagonal matrix as the matrix containing the exponentials of the diagonal entries, i.e.

$$ e^{\Lambda} = \left( \begin{array}{cccc} e^{\lambda_1} & \dots & 0 & 0 \\ 0 & e^{\lambda_2} & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & e^{\lambda_n} \end{array} \right) $$

Now we can consider more general matrices. For simplicity, I'll assume that $A$ is diagonalizable. Then you can define the matrix exponential as

$$ e^A = P e^\Lambda P^{-1} $$

where $A = P \Lambda P^{-1}$. If the matrix isn't diagonalizable, one needs the Jordan form of the matrix, which leads to some interesting things that I'll let you discover!