Are complex numbers real matrices? 90 degree rotation matrices are imaginary units

Antonio Sala, UPV

Difficulty: * ,       Relevance: ,      Duration: 23:17

Summary:

This video examines how the ‘field of complex numbers’ can be formally represented using square matrices.

Through various examples, it demonstrates that an identity matrix and a specific 90-degree rotation matrix act as the real and imaginary units, respectively. Operations such as multiplication, transposition, and inversion validate that these structures maintain the fundamental properties of complex numbers. Furthermore, the application of Euler’s formula (exponentials and their derivatives) in this context is explored, confirming that matrix operations exactly mimic complex algebra.

Last, it is briefly mentioned that there are an infinite number of matrices that can serve as imaginary units, which will give rise to the quaternion algebra and other quantum-mechanical mathematical frameworks (out of the scope of this video).

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.