Verifying matrix exponential identity

Question

I wish to verify the following matrix exponential identity using Mathematica:

$$ \mathbb{e}^{i.x.\hat{n}.\sigma}= \cos{x}.\mathbb{I}+ i\sin{x}(\hat{n}.\sigma) $$

where $\mathbb{I}$ is the $2\times2$ identity matrix, $i$ is the imaginary unit, $\hat{n}$ is an arbitrary $3$-dimensional unit vector, $x$ is an arbitrary real number, and $\sigma$ is the $3$ component Pauli vector(whose components each contain one of the Pauli matrices).

I have tried the following:

 A = Array[PauliMatrix, 3]

 n = Normalize[{a,b,c}]

 MatrixExp[I*x*(n . A)] == Cos[x]*IdentityMatrix[2] + I*Sin[x]*(n . A)

but, to no avail. Can someone give any suggestions?

Answer 1

I think the easiest and fastest way is to help Mathematica. That is to use a FullSimplify with some Assumptions at the level of (n . A). Then, it's really fast. What this does is getting rid of the Abs in the expressions that were present without the assumptions.

MatrixExp[
    I*x*FullSimplify[(n . A), 
      Assumptions -> {a >= 0 && b >= 0 && c >= 0}]] == 
   Cos[x]*IdentityMatrix[2] + 
    I*Sin[x]*
     FullSimplify[(n . A), 
      Assumptions -> {a >= 0 && b >= 0 && c >= 0}] // 
  Factor // Simplify

and another case

MatrixExp[
    I*x*FullSimplify[(n . A), 
      Assumptions -> {a < 0 && b < 0 && c < 0}]] == 
   Cos[x]*IdentityMatrix[2] + 
    I*Sin[x]*
     FullSimplify[(n . A), Assumptions -> {a < 0 && b < 0 && c < 0}] //
   Factor // Simplify

both give True