*I know the sentence that for normal matrices, the eigenvectors, of different eigenvalues, are orthogonal.
Yet, I also know that every normal matrix is unitary diagonalaizable.
Also, the columns of the diagonalazing matrix is built from the eigenvectors and for a unitary matrix, the coulmns are orthogonal.
Therefore, it seems to me that all the eigenvectors of a normal matrix, and it doesnt matter if its from the same or different eigenvalue, should be orthogonal to each other.
Yet, i only familiar with the orthogonality of eigenvectors in normal matrix for distinct eigenvalues.
But, according to the explanation i gave, why not all the eigenvectors of a normal matrix are orthogonal to each other, regardeless of the eigenvalue?