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In mathematics, a '''Hermitian matrix''' (or '''self-adjoint matrix''') is a [[square matrix]] with [[complex number|complex]] entries that is equal to its own [[conjugate transpose]]—that is, the element in the {{mvar|i}}-th row and {{mvar|j}}-th column is equal to the [[complex conjugate]] of the element in the {{mvar|j}}-th row and {{mvar|i}}-th column, for all indices {{mvar|i}} and {{mvar|j}}:
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:<math>a_{ij} = \overline{a_{ji}}</math> or <math>A^T = \overline A</math>, in matrix form.
 
Hermitian matrices can be understood as the complex extension of real [[symmetric matrix|symmetric matrices]].
 
If the conjugate transpose of a matrix <math>A</math> is denoted by <math>A^\dagger</math>, then the Hermitian property can be written concisely as
 
:<math> A = A^\dagger\,.</math>
 
Note that asterisk <math>A^*</math> is commonly used to represent <math>A^\dagger</math> although some authors have been known to use it to represent <math>\overline A</math> which can create confusion.
 
Hermitian matrices are named after [[Charles Hermite]], who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having [[Eigenvalues and eigenvectors|eigenvalues]] always real.
 
== Examples ==
See the following example:
 
:<math>
\begin{bmatrix}
2 & 2+i & 4 \\
2-i & 3 & i \\
4 & -i & 1 \\
\end{bmatrix}
</math>
 
The diagonal elements must be [[real number|real]], as they must be their own complex conjugate.  
 
Well-known families of [[Pauli matrices]], [[Gell-Mann matrices]] and their generalizations are Hermitian. 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 |authorlink=Theodore Frankel |year=2004 |publisher=[[Cambridge University Press]] |isbn=0-521-53927-7 |page=652 |url=http://books.google.ru/books?id=DUnjs6nEn8wC&lpg=PA652&dq=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652#v=onepage&q&f=false }}
</ref><ref>
[http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf Physics 125 Course Notes] at [[California Institute of Technology]]
</ref> which results in ''skew-Hermitian'' matrices (see [[#facts|below]]).
 
== Properties ==
 
* The entries on the [[main diagonal]] (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian [[if and only if]] it is a [[symmetric matrix]], i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.
 
* Every Hermitian matrix is a [[normal matrix]].
 
* The finite-dimensional [[spectral theorem]] says that any Hermitian matrix can be [[diagonalizable matrix|diagonalized]] by a [[unitary matrix]], and that the resulting diagonal matrix has only real entries. This implies that all [[eigenvectors|eigenvalue]]s of a Hermitian matrix {{mvar|A}} are real, and that {{mvar|A}} has {{mvar|n}} linearly independent [[eigenvector]]s. Moreover, it is possible to find an [[orthonormal basis]] of {{math|'''C'''<sup>''n''</sup>}} consisting of {{mvar|n}} eigenvectors of {{mvar|A}}.
 
* The sum of any two Hermitian matrices is Hermitian, and the [[inverse matrix|inverse]] of an invertible Hermitian matrix is Hermitian as well. However, the [[matrix multiplication|product]] of two Hermitian matrices {{mvar|A}} and {{mvar|B}} is Hermitian if and only if {{math|1=''AB'' = ''BA''}}. Thus {{math|''A''<sup>''n''</sup>}} is Hermitian if {{mvar|A}} is Hermitian and {{mvar|n}} is an integer.
 
* The Hermitian complex {{mvar|n}}-by-{{mvar|n}} matrices do not form a [[vector space]] over the [[complex number]]s, since the identity matrix {{math|''I''<sub>n</sub>}} is Hermitian, but {{math|''i'' ''I''<sub>n</sub>}} is not. However the complex Hermitian matrices ''do'' form a vector space over the [[real numbers]] {{math|'''R'''}}.  In the {{math|2''n''<sup>2</sup>}}-[[dimension of a vector space|dimensional]] vector space of complex {{math|''n'' × ''n''}} matrices over {{math|'''R'''}}, the complex Hermitian matrices form a subspace of dimension {{math|''n''<sup>2</sup>}}. If {{math|''E''<sub>''jk''</sub>}} denotes the {{mvar|n}}-by-{{mvar|n}} matrix with a 1 in the {{math|''j'',''k''}} position and zeros elsewhere, a basis can be described as follows:
::<math>\; E_{jj}</math> for <math>1\leq j\leq n</math> ({{mvar|n}} matrices)
:together with the set of matrices of the form
::<math>\; E_{jk}+E_{kj}</math> for <math>1\leq j<k\leq n </math>    ({{sfrac|''n''<sup>2</sup> − ''n''|2}} matrices)
:and the matrices
::<math>\; i(E_{jk}-E_{kj})</math> for <math>1\leq j<k\leq n </math>    ({{sfrac|''n''<sup>2</sup> − ''n''|2}} matrices)
:where <math>i</math> denotes the complex number <math>\sqrt{-1}</math>, known as the [[imaginary unit]].
 
* If {{mvar|n}} orthonormal eigenvectors <math>u_1,\dots,u_n</math> of a Hermitian matrix are chosen and written as the columns of the matrix {{mvar|U}}, then one [[Eigendecomposition of a matrix|eigendecomposition]] of {{mvar|A}} is <math> A = U \Lambda U^\dagger</math> where <math> U U^\dagger = I=U^\dagger U</math> and therefore
::<math> A = \sum _j \lambda_j u_j u_j ^\dagger </math>,
:where <math>\lambda_j</math> are the eigenvalues on the diagonal of the diagonal matrix <math>\; \Lambda </math>.
 
== Further properties ==
{{anchor|facts}}Additional facts related to Hermitian matrices include:
* The sum of a square matrix and its conjugate transpose <math>(C + C^{\dagger})</math> is Hermitian.
* The difference of a square matrix and its conjugate transpose <math>(C - C^{\dagger})</math> is [[skew-Hermitian matrix|skew-Hermitian]] (also called antihermitian). This implies that [[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}}:
::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^{\dagger}) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^{\dagger}).</math>
* The determinant of a Hermitian matrix is real:
:: Proof: <math> \det(A) = \det(A^\mathrm{T})\quad \Rightarrow \quad \det(A^\dagger) = \det(A)^* </math>
:: Therefore if <math>A=A^\dagger\quad \Rightarrow \quad \det(A) = \det(A)^*.</math>
:(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
 
==See also==
*[[Skew-Hermitian matrix]] (anti-Hermitian matrix)
*[[Haynsworth inertia additivity formula]]
*[[Hermitian form]]
*[[Self-adjoint operator]]
*[[Unitary matrix]]
 
==References==
{{reflist}}
 
==External links==
*{{springer|title=Hermitian matrix|id=p/h047070}}
*[http://people.ofset.org/~ckhung/b/la/hermitian.en.php Visualizing Hermitian Matrix as An Ellipse with Dr. Geo], by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
*{{MathPages|id=home/kmath306/kmath306|title=Hermitian Matrices}}
 
[[Category:Matrices]]

Latest revision as of 05:51, 13 October 2014

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