Non-Symmetric Matrix: Real Eigenvalues Examples

[LagrangeEuler]

Do you know any example of matrix which is not symmetric and has real eigenvalues?

 

[maajdl]

Any lower (or upper) triangular matrix with a real diagonal would be an example.
Starting from this example, you could generate as many others as you want by applying a change of basis that would render the matrix non-triangular, but keep its eigenvalues the same.