Editing Hermitian matrix (section)

==Decomposition into Hermitian and skew-Hermitian matrices==

{{anchor|facts}}Additional facts related to Hermitian matrices include:
* The sum of a square matrix and its conjugate transpose <math>\left(A + A^\mathsf{H}\right)</math> is Hermitian.
* The difference of a square matrix and its conjugate transpose <math>\left(A - A^\mathsf{H}\right)</math> is [[skew-Hermitian matrix|skew-Hermitian]] (also called antihermitian). This implies that the [[commutator]] of two Hermitian matrices is skew-Hermitian.
* An arbitrary square matrix {{mvar|C}} can be written as the sum of a Hermitian matrix {{mvar|A}} and a skew-Hermitian matrix {{mvar|B}}. This is known as the Toeplitz decomposition of {{mvar|C}}.<ref name="HornJohnson">{{cite book |title=Matrix Analysis, second edition |first1=Roger A. |last1=Horn |first2=Charles R. |last2=Johnson |isbn=9780521839402 |publisher=Cambridge University Press|year=2013}}</ref>{{rp|227}} <math display="block">C = A + B \quad\text{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\text{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right)</math>