Editing Hermitian matrix (section)

==Examples and solutions==

In this section, the conjugate transpose of matrix <math> A </math> is denoted as <math> A^\mathsf{H} ,</math> the transpose of matrix <math> A </math> is denoted as <math> A^\mathsf{T} </math> and conjugate of matrix <math> A </math> is denoted as <math> \overline{A} .</math>

See the following example:

<math display=block>\begin{bmatrix}
  0     & a - ib & c-id \\
  a+ib  & 1     & m-in \\
  c+id     &   m+in & 2
\end{bmatrix}</math>

The diagonal elements must be [[real number|real]], as they must be their own complex conjugate.

Well-known families of Hermitian matrices include the [[Pauli matrices]], the [[Gell-Mann matrices]] and their generalizations. In [[theoretical physics]] such Hermitian matrices are often multiplied by [[imaginary number|imaginary]] coefficients,<ref>
{{cite book |title=The Geometry of Physics: an introduction |last=Frankel |first=Theodore |author-link=Theodore Frankel |year=2004 |publisher=[[Cambridge University Press]] |isbn=0-521-53927-7 |page=652 |url=https://books.google.com/books?id=DUnjs6nEn8wC&q=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652 }}
</ref><ref>[http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf Physics 125 Course Notes] {{Webarchive|url=https://web.archive.org/web/20220307172254/http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf |date=2022-03-07 }} at [[California Institute of Technology]]</ref> which results in [[Skew-Hermitian matrix|skew-Hermitian matrices]].

Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <math> A </math> equals the [[matrix multiplication|product of a matrix]] with its conjugate transpose, that is, <math> A = BB^\mathsf{H} ,</math> then <math> A </math> is a Hermitian [[positive semi-definite matrix]]. Furthermore, if <math> B </math> is row full-rank, then <math> A </math> is positive definite.