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1) When we want to find the unitary matrix that block-diagonalizes a certain matrix through a similarity transformation, we should find the eigenvectors of that matrix and stick them together to get a square matrix. But this process of sticking two column matrices together doesn't seem rigorous to me. Is there a rigorous way of doing this?

2)Consider the group of rotations around an axis. It has two irreducible representations, [itex] e^{\pm i \varphi} [/itex] on the vector space of complex numbers. But we also have the reducible representation of [itex] \mathbb R ^2 [/itex] with Rotation matrices!

a) They both seem to be two dimensional. So sub-representations can have the same dimension of the bigger representation? Seems strange!

b) For [itex] e^{i\varphi}[/itex], we write the vectors as [itex] z=x+iy=\rho e^{i\alpha} [/itex]. Should we write the vectors differently when we consider [itex] e^{-i\varphi} [/itex]?

c)can we say the group of rotations around an axis has a left-handed and a right-handed irreducible representation?

Thanks