- Hermitian matrix
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**Hermitian matrix**(or**self-adjoint matrix**) is asquare matrix with complex entries which is equal to its ownconjugate transpose — that is, the element in the "i"th row and "j"th column is equal to thecomplex conjugate of the element in the "j"th row and "i"th column, for all indices "i" and "j"::$a\_\{i,j\}\; =\; overline\{a\_\{j,i.$

If the conjugate transpose of a matrix "A" is denoted by $A^*$, then this can concisely be written as

:$A\; =\; A^*.\; ,$

For example,

:$egin\{bmatrix\}32+i\backslash 2-i1end\{bmatrix\}$

is a Hermitian matrix.

**Properties**The entries on the

main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real. A matrix that has only real entries is Hermitian if and only if it is asymmetric 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 normal, and the finite-dimensional

spectral theorem applies. It says that any Hermitian matrix can be diagonalized by aunitary matrix , and that the resulting diagonal matrix has only real entries. This means that all eigenvalues of a Hermitian matrix are real, and, moreover,eigenvector s with distinct eigenvalues are orthogonal. It is possible to find anorthonormal basis of**C**^{"n"}consisting only of eigenvectors.The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices "A" and "B" will only be Hermitian if they commute, i.e., if "AB" = "BA". Thus "A"

^{"n"}is Hermitian if "A" is Hermitian and "n" is a positive integer.The Hermitian "n"-by-"n" matrices form a

vector space over thereal number s (but not over the complex numbers). The dimension of this space is "n"^{2}(one degree of freedom per main diagonal element, and two degrees of freedom per element above the main diagonal).The eigenvectors of an Hermitian matrix are orthogonal, i.e. its eigendecomposition is $A\; =\; U\; Sigma\; U^dagger$ where $U\; U^dagger\; =\; I$. Since right- and left- inverse are the same, we also have $U^dagger\; U\; =\; I$, and therefore $A\; =\; sum\; \_i\; sigma\_i\; u\_i\; u\_i\; ^dagger$ (where $sigma\_i$ are the eigenvalues and $u\_i$ the eigenvectors.

If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite; if they are all non-negative, then the matrix is positive semidefinite.

Additional properties of Hermitian matrices include:

* The sum of a square matrix and its conjugate transpose $(C\; +\; C^\{dagger\})$ is Hermitian.

* The difference of a square matrix and its conjugate transpose $(C\; -\; C^\{dagger\})$ isskew-Hermitian (also called antihermitian).

* An arbitrary square matrix "C" can be written as the sum of a Hermitian matrix "A" and a skew-Hermitian matrix "B":::$C\; =\; A+B\; quadmbox\{with\}quad\; A\; =\; frac\{1\}\{2\}(C\; +\; C^\{dagger\})\; quadmbox\{and\}quad\; B\; =\; frac\{1\}\{2\}(C\; -\; C^\{dagger\}).$**ee also***

Hermitian form

*Hermitian operator **External links*** [

*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.

* [*http://www.mathpages.com/home/kmath306/kmath306.htm Hermitian Matrices*] at MathPages

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**Hermitian matrix**— ermitinė matrica statusas T sritis fizika atitikmenys: angl. Hermitian matrix; self adjoint matrix vok. Hermite Matrix, f; hermitesche Matrix, f; selbstadjungierte Matrix, f rus. самосопряжённая матрица, f; эрмитова матрица, f pranc. matrice… … Fizikos terminų žodynas**hermitian matrix**— hər¦mishən , erˈmēshən noun Usage: usually capitalized H Etymology: Charles Hermite died 1901 French mathematician + English ian : a square matrix having the property that each pair of elements comprised of one in the ith row and jth column and… … Useful english dictionary**Hermitian matrix**— Math. a matrix, whose entries are complex numbers, equal to the transpose of the matrix whose entries are the conjugates of the entries of the given matrix. [1925 30; after C. HERMITE; see IAN] * * * … Universalium**Hermitian matrix**— noun Etymology: Charles Hermite died 1901 French mathematician Date: 1935 a square matrix having the property that each pair of elements in the ith row and jth column and in the jth row and ith column are conjugate complex numbers … New Collegiate Dictionary**Skew-Hermitian matrix**— In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew Hermitian or antihermitian if its conjugate transpose A * is also its negative.… … Wikipedia**Moore determinant of a Hermitian matrix**— Not to be confused with Moore determinant over a finite field. In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore (1922). See also Dieudonné determinant… … Wikipedia**Hermitian variety**— Hermitian varieties are in a sense a generalisation of quadrics, and occur naturally in the theory of polarities.DefinitionLet K be a field with an involutive automorphism heta. Let n be an integer geq 1 and V be an (n+1) dimensional vectorspace… … Wikipedia**Matrix mechanics**— Quantum mechanics Uncertainty principle … Wikipedia**Hermitian manifold**— In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define … Wikipedia**Hermitian**— A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint *Hermitian connection *Hermitian form *Hermitian function *Hermitian hat wavelet *Hermitian kernel *Hermitian manifold/structure… … Wikipedia