Editing Eigenvalue algorithm (section)

===Normal, Hermitian, and real-symmetric matrices===
{{main|Adjoint matrix|Normal matrix|Hermitian matrix}}
The [[conjugate transpose|adjoint]] {{math|''M''<sup>*</sup>}} of a complex matrix {{math|''M''}} is the transpose of the conjugate of {{math|''M''}}: {{math|1=''M'' <sup>*</sup> = {{overline|''M''}} <sup>T</sup>}}. A square matrix {{math|''A''}} is called ''[[Normal matrix|normal]]'' if it commutes with its adjoint: {{math|1=''A''<sup>*</sup>''A'' = ''AA''<sup>*</sup>}}. It is called ''[[Hermitian matrix|Hermitian]]'' if it is equal to its adjoint: {{math|1=''A''<sup>*</sup> = ''A''}}. All Hermitian matrices are normal. If {{math|''A''}} has only real elements, then the adjoint is just the transpose, and {{math|''A''}} is Hermitian if and only if it is [[symmetric matrix|symmetric]]. When applied to column vectors, the adjoint can be used to define the canonical inner product on {{math|'''C'''<sup>''n''</sup>}}: {{math|1='''w''' ⋅ '''v''' = '''w'''<sup>*</sup> '''v'''}}.<ref group="note">This ordering of the inner product (with the conjugate-linear position on the left), is preferred by physicists. Algebraists often place the conjugate-linear position on the right: {{math|1='''w''' ⋅ '''v''' = '''v'''<sup>*</sup> '''w'''}}.</ref> Normal, Hermitian, and real-symmetric matrices have several useful properties:
* Every generalized eigenvector of a normal matrix is an ordinary eigenvector.
* Any normal matrix is similar to a diagonal matrix, since its Jordan normal form is diagonal.
* Eigenvectors of distinct eigenvalues of a normal matrix are orthogonal. 
* The null space and the image (or column space) of a normal matrix are orthogonal to each other. 
* For any normal matrix {{math|''A''}}, {{math|'''C'''<sup>''n''</sup>}} has an orthonormal basis consisting of eigenvectors of {{math|''A''}}. The corresponding matrix of eigenvectors is [[Unitary matrix|unitary]].
* The eigenvalues of a Hermitian matrix are real, since {{math|1=({{overline|''λ''}} − ''λ'')'''v''' = (''A''<sup>*</sup> − ''A'')'''v''' = (''A'' − ''A'')'''v''' = 0}} for a non-zero eigenvector {{math|'''v'''}}.
* If {{math|''A''}} is real, there is an orthonormal basis for {{math|'''R'''<sup>''n''</sup>}} consisting of eigenvectors of {{math|''A''}} if and only if {{math|''A''}} is symmetric.

It is possible for a real or complex matrix to have all real eigenvalues without being Hermitian. For example, a real [[triangular matrix]] has its eigenvalues along its diagonal, but in general is not symmetric.