en>LivingInterested: /* Indeterminacy and incompleteness */ "(local)" added, since alternate "realities", as consciousness, has non local as well as local characteristics.

2014-10-13T04:51:16Z

Indeterminacy and incompleteness: "(local)" added, since alternate "realities", as consciousness, has non local as well as local characteristics.

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en>Egsan Bacon: Repairing links to disambiguation pages - You can help!

2013-10-31T16:25:36Z

Repairing links to disambiguation pages - You can help!

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	~~Valuer Alderman from Saint~~-~~Pierre~~-de-l'~~Ile~~-d'~~Orleans~~, ~~really likes crosswords~~, ~~simpsons tapped out hack~~ and ~~candle making~~. ~~Will soon undertake~~ a ~~contiki trip~~ that ~~may include visiting~~ the ~~St Mary~~'s ~~Cathedral~~ and ~~St Michael~~'~~s Church at Hildesheim~~.<br><br>~~Review my site~~; [~~https~~://~~www~~.~~facebook~~.~~com~~/~~TheSimpsonsTappedOutHack2014 simpsons tapped out hack android~~]		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}}:

			:<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]]

en>Maurice Carbonaro: Hyperlinked "(...) indeterminacy (...)" with the "Uncertainty principle" article. Please feel free to undo... motivating, if possible, your decision. Thanks.

2012-08-09T08:16:57Z

Hyperlinked "(...) indeterminacy (...)" with the "Uncertainty principle" article. Please feel free to undo... motivating, if possible, your decision. Thanks.

New page

Valuer Alderman from Saint-Pierre-de-l'Ile-d'Orleans, really likes crosswords, simpsons tapped out hack and candle making. Will soon undertake a contiki trip that may include visiting the St Mary's Cathedral and St Michael's Church at Hildesheim.<br><br>Review my site; [https://www.facebook.com/TheSimpsonsTappedOutHack2014 simpsons tapped out hack android]

Quantum indeterminacy - Revision history

en>LivingInterested: /* Indeterminacy and incompleteness */ "(local)" added, since alternate "realities", as consciousness, has non local as well as local characteristics.

en>Egsan Bacon: Repairing links to disambiguation pages - You can help!

en>Maurice Carbonaro: Hyperlinked "(...) indeterminacy (...)" with the "Uncertainty principle" article. Please feel free to undo... motivating, if possible, your decision. Thanks.