Editing Normal matrix (section)

==Special cases==
Among complex matrices, all [[unitary matrix|unitary]], [[Hermitian matrix|Hermitian]], and [[skew-Hermitian matrix|skew-Hermitian]] matrices are normal, with all eigenvalues being unit modulus, real, and imaginary, respectively. Likewise, among real matrices, all [[orthogonal matrix|orthogonal]], [[symmetric matrix|symmetric]], and [[skew-symmetric matrix|skew-symmetric]] matrices are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it is ''not'' the case that all normal matrices are either unitary or (skew-)Hermitian, as their eigenvalues can be any complex number, in general.  For example,
<math display="block">A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}</math>
is neither unitary, Hermitian, nor skew-Hermitian, because its eigenvalues are <math>2, (1\pm i\sqrt{3})/2</math>; yet it is normal because 
<math display="block">AA^* = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} = A^*A.</math><!--
For the curious, the four classes
     \begin{bmatrix} a & b & 0 \\ 0 & a &  b \\  b & 0 & a \end{bmatrix}
     \begin{bmatrix} a & b & 0 \\ 0 & a & -b \\  b & 0 & a \end{bmatrix}
     \begin{bmatrix} a & b & 0 \\ 0 & a &  b \\ -b & 0 & a \end{bmatrix}
     \begin{bmatrix} a & b & 0 \\ 0 & a & -b \\ -b & 0 & a \end{bmatrix}
     are neither unitary nor skew-Hermitian for all non-zero real a and b. There are more 3×3 examples, but among 2×2 matrices, there are only ones that are multiples of unitary matrices.
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