Theorem 14.14 A matrix can be diagonalized by a unitary similarity transformation, if and only if it is a normal matrix.
Thus, Theorem 14.13 is naturally generalized so that it can be stated as follows:
Two normal matrices B and C commute if and only if there exists a complete
orthonormal set of common eigenvectors.
In Sects. 14.1 and 14.3, we mentioned the spectral decomposition. There, we
showed a special case where projection operators commute with one another; see
(14.23). Thus, in light of Theorem 14.13, those projection operators can be diagonalized at once to be expressed as, e.g., (14.72). This is a conspicuous feature of the
projection operators.
References
1. Hassani S (2006) Mathematical physics. Springer, New York
2. Byron FW Jr, Fuller RW (1992) Mathematics of classical and quantum physics. Dover,
New York
3. Mirsky L (1990) An introduction to linear algebra. Dover, New York
4. Satake I-O (1975) Linear algebra (Pure and applied mathematics). Marcel Dekker, New York
5. Satake I (1974) Linear algebra. Shokabo, Tokyo. (in Japanese)
580
14 Hermitian Operators and Unitary Operators