The real spectral theorem asserts that any symmetric matrix can be decomposed into a composition of rotations, reflections and scaling. Why can't a non-symmetric matrix be represented as such? Are there other types of operations other than rotations, reflections and scaling that explain why non-symmetric matrices are left out of this theorem? I understand that the singular value decomposition says that you can decompose a matrix into 3 other matrices - but the matrices are complex and I don't know if you can interpret a complex matrix as a rotation etc. I apologise if this is a silly question and please let me know if the question requires further clarification.