Hermitian matrix

2020-05-29T19:00:54Z

FixingPhysics: Fixed sloppy description where diagonal entries were mistakenly allowed to be complex.

A '''Hermitian matrix''' is one that satisfies <math>M=M^\dagger</math>, where <math>M^\dagger</math> is the Hermitian conjugate of <math>M</math> (i.e., the matrix formed by transposing <math>M</math> and taking the [[complex conjugate]] of each element). As an example, the most general 2x2 Hermitian [[matrix]] has the form

<math>
\begin{pmatrix}
a & b \\
b^* & c
\end{pmatrix}
</math>

for any complex number <math>b</math> and real numbers <math>a</math> and <math>c</math>. In the case where all elements of the matrix are real, a Hermitian matrix becomes symmetric (as Hermitian conjugation then becomes equivalent to transposition).

==Properties of Hermitian matrices==
* The [[eigenvalue]]s are all real.
* The [[eigenvector]]s corresponding to different eigenvalues are orthogonal.

Because of these properties, Hermitian matrices have important applications in [[quantum mechanics]].
[[Category:Linear Algebra]]

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Hermitian matrix