Special Cases
Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal.
However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian. As an example, the matrix
is normal because
The matrix A is neither unitary, Hermitian, nor skew-Hermitian.
The sum or product of two normal matrices is not necessarily normal. If they commute, however, then this is true.
If A is both a triangular matrix and a normal matrix, then A is diagonal. This can be seen by looking at the diagonal entries of A*A and AA*, where A is a normal, triangular matrix.
Read more about this topic: Normal Matrix
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