I am revising Linear Algebra and trying to put together a rule how to decide whether a matrix is diagonaliseable. (a short and simple rule that can be referred to immediately when solving a problem)

This is what I came up based on my experience with solving questions and comparing to model answers.

Real matrices.

Only symmetric matrices are orthogonally diagonaliseable - by any orthogonal matrix whose column vectors are eigenvectors of the original (symmetric) matrix.

No need for Gram-Schmidt orthonormalisation process here as it is not required for this vectors to be orthonormal.

Complex matrices.

Only normal matrices are unitarily diagonaliseable - by any unitary matrix whose column vectors are eigenvectors of A of the original (normal) matrix.

To unitarily diagonalise a normal matrix, need to find orthonormal basis (ie apply Gram-Schmidt process if necessary and have normalised vectors in the basis)

Is this correct?