Exponential of Paulis

Basic Quantum Mechanics

Show that exponential of a Pauli matrix $A$ can be simplified as follows:

$$ \exp(i A \theta) = \cos(\theta) I + i \sin(\theta) A $$

Step 1

It's easy to think of exponentiating numbers but how should we think about the exponential of a matrix. Let's say I have a matrix called $B$:

$$ B = \begin{pmatrix} b_{11} & \cdots & b_{1m} \\ b_{21} & \cdots & b_{2m} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nm} \\ \end{pmatrix} $$

And let's assume that when I exponentiate this matrix I end up with another matrix. How would you define $\exp(B)$?

Replace each element of the matrix, $b_{ij}$, with its exponentials that is $\exp(b_{ij})$. $$ \exp(B) = \begin{pmatrix} \exp(b_{11}) & \cdots & \exp(b_{1m}) \\ \exp(b_{21}) & \cdots & \exp(b_{2m}) \\ \vdots & \ddots & \vdots \\ \exp(b_{n1}) & \cdots & \exp(b_{nm}) \\ \end{pmatrix} $$

Define the exponential function using Taylor series.

Use Euler's formula for complex numbers.