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Ερμιτιανή Μήτρα

- Ένα είδος Μήτρας.

## ΕτυμολογίαEdit

Η ονομασία "Ερμιτιανός" σχετίζεται ετυμολογικά με το όνομα "Hermite" who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real..

## ΕισαγωγήEdit

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

$a_{ij} = \overline{a_{ji}}\,.$

If the conjugate transpose of a matrix $A$ is denoted by $A^\dagger$, then the Hermitian property can be written concisely as

$A = A^\dagger\,.$

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

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 always real.

## Examples Edit

For example,

$\begin{bmatrix}3&2+i\\ 2-i&1\end{bmatrix}.$

Well-known families of Pauli matrices, Gell-Mann matrices and various generalizations are Hermitian. In theoretical physics such Hermitian matrices usually are multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below).

## Properties Edit

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 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, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries.

This implies that all eigenvalues of a Hermitian matrix A are real, and that A has n linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of Cn consisting of n eigenvectors of A.

• 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 is Hermitian if they commute, i.e., if AB = BA. Thus An is Hermitian if A is Hermitian and n is an integer.

The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, since the identity matrix $I_n$ is Hermitian, but $i(I_n)$ is not. However the complex Hermitian matrices do form a vector space over the real numbers. In the 2n2 R dimensional vector space of complex n×n matrices, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis can be described as follows:

$\; E_{jj}$ for $1\leq j\leq n$ (n matrices)

together with the set of matrices of the form

$\; E_{jk}+E_{kj}$ for $1\leq j<k\leq n$ ((n2n)/2 matrices)

and the matrices

$\; i(E_{jk}-E_{kj})$ for $1\leq j<k\leq n$ ((n2n)/2 matrices)

where $i$ denotes the complex number $\sqrt{-1}$, known as the imaginary unit.

If n orthonormal eigenvectors $u_1,\dots,u_n$ of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is $A = U \Lambda U^\dagger$ where $U U^\dagger = I=U^\dagger U$ and therefore

$A = \sum _j \lambda_j u_j u_j ^\dagger$,

where $\lambda_j$ are the eigenvalues on the diagonal of the diagonal matrix $\; \Lambda$.

Πρότυπο:AnchorAdditional facts related to 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 $(C - C^{\dagger})$ is skew-Hermitian (also called antihermitian).
• This implies that commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:
$C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^{\dagger}) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^{\dagger}).$
• The determinant of a Hermitian matrix is real:
Proof: $\det(A) = \det(A^\mathrm{T})\quad \Rightarrow \quad \det(A^\dagger) = \det(A)^*$
Therefore if $A=A^\dagger\quad \Rightarrow \quad \det(A) = \det(A)^*.$
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

### Pauli spin matricesEdit

Any 2x2 Hermitian matrix may be written as a linear combination of the 2×2 identity matrix and the three Pauli spin matrices.

These matrices have use in quantum mechanics.

The four matrices form an orthogonal basis for the 4-dimensional vector space of 2x2 Hermitian matrices.

$\mathbf{I}=\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \sigma_{x}=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix},\quad \sigma_{y}=\begin{pmatrix}0 & -\mathit{i} \\ \mathit{i} & 0 \end{pmatrix},\quad \sigma_{z}=\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$

An arbitrary 2×2 Hermitian matrix A is written thus,

$\mathbf{A} = a\mathbf{I} + b\sigma_{x} + c\sigma_{y} + d\sigma_{z}$
$A = \begin{pmatrix}a & 0 \\ 0 & a \end{pmatrix} + \begin{pmatrix}0 & b \\ b & 0 \end{pmatrix} + \begin{pmatrix}0 & -\mathit{i}c \\ \mathit{i}c & 0 \end{pmatrix} + \begin{pmatrix}d & 0 \\ 0 & -d \end{pmatrix}$
$A =\begin{pmatrix}a+d & b-\mathit{i}c \\ b+\mathit{i}c & a-d \end{pmatrix}$

## Skew-Hermitian MatricesEdit

A skew-Hermitian matrix is one which is equal to the negative of its Hermitian adjoint:

$\mathbf{A=-A^*}$

For instance, (a, b, c, d, e, f, g, h, and k are real),

$\mathbf{A}=\begin{pmatrix} \mathit{i}a & -b+\mathit{i}c & -e+\mathit{i}f \\ b+\mathit{i}c & \mathit{i}d & -h+\mathit{i}k \\ e+\mathit{i}f & h+\mathit{i}k & \mathit{i}g \end{pmatrix}$

is a skew-Hermitian matrix.

Clearly, the entries on the main diagonal are purely imaginary.

## ΕφαρμογήEdit

Σε κάθε Φυσικό Μέγεθος αντιστοιχεί ένας κατάλληλος Ερμιτιανός Τελεστής που αναπαρίσταται από μία Ερμιτιανή Μήτρα, του οποίου οι ιδιοτιμές είναι τα μοναδικά δυνατά εξαγόμενα μιας μέτρησης.

## ΙστογραφίαEdit

Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Επίσης,
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν

>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)