Editing Normal matrix (section)

==Normal matrix analogy==
{{confusing|date=October 2023|analogies in the below list}}
It is occasionally useful (but sometimes misleading) to think of the relationships of special kinds of normal matrices as analogous to the relationships of the corresponding type of complex numbers of which their eigenvalues are composed.  This is because any function (that can be expressed as a power series) of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition <math>VD V^*</math> is <math>VD^*V^*</math>, where <math>D</math> is the diagonal matrix of eigenvalues.  Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on each corresponding pair of eigenvalues.
* The [[conjugate transpose]] is analogous to the [[complex conjugate]].
* [[Unitary matrix|Unitary matrices]] are analogous to [[complex number]]s on the [[unit circle]].
* [[Hermitian matrix|Hermitian matrices]] are analogous to [[real number]]s.
* Hermitian [[positive-definite matrix|positive definite matrices]] are analogous to [[positive real numbers]].
* [[Skew-Hermitian matrix|Skew Hermitian matrices]] are analogous to purely [[imaginary number]]s.
* [[Inverse matrix|Invertible matrices]] are analogous to non-zero [[complex number]]s.
* The inverse of a matrix has each eigenvalue inverted.
* A uniform [[Scaling_(geometry)|scaling matrix]] is analogous to a constant number.
* In particular, the [[zero matrix|zero matrix]] is analogous to 0, and
* the [[identity matrix|identity]] matrix is analogous to 1.
* An [[idempotent matrix]] is an orthogonal projection with each eigenvalue either 0 or 1.
* A normal [[Involutory_matrix|involution]] has eigenvalues <math>\pm 1</math>.

As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mapping 
<math display="block">a + bi \mapsto \begin{bmatrix} a & b \\ -b & a \end{bmatrix} = a\,  \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b\, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\,.</math>
which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.