I have to find out a normal transformation that is neither hermitian nor unitary. http://en.wikipedia.org/wiki/Normal_matrix gives me the answer. However, I would like to know how to find it out mathematically, not just guess and test. Can I go from the properties of eigenvalues of hermitian and unitary to get the answer? For example: eigenvalues of a hemitian must be real, then I choose (i,-i,0) as eigenvalues of the required matrix. Those eigenvalues satisfy the condition that the required matrix is not unitary whose eigenvalues are |1|. From those, I have characteristic equation has the form of x(x^2+1)= 0 which leads to the form of the required matrix has the trace =0 and det = 0. Can I do it?
The answer to your (imprecise) question lies in the Spectral theorem for normal matrices: normal matrices are precisely those that are unitarily diagonalizable. Hermitian and unitary matrices are special cases: hermitian matrices are normal with real eigenvalues, while unitary matrices are normal with complex eigenvalues of modulus one.
Therefore to answer your question, you should look for some matrix with complex non-real and non-unitary eigenvalues.