Normal matrix
In mathematics, a complex square matrix A is normal if
where A∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.
A real square matrix A satisfies A∗ = AT, and is therefore normal if ATA = AAT.
A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A∗A = AA∗ is diagonalizable.
The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.
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. For example,
is neither unitary, Hermitian, nor skew-Hermitian, yet it is normal because
Let A be a normal upper triangular matrix. Since (A∗A)ii = (AA∗)ii, one has ⟨ei, A*Aei⟩= ⟨ei, AA*ei⟩ i.e. the first row must have the same norm as the first column:
...
Wikipedia