- Ένα είδος Μήτρας.
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..
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:
If the conjugate transpose of a matrix is denoted by , then the Hermitian property can be written concisely as
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
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, which results in skew-Hermitian matrices (see below).
(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 is Hermitian, but 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:
- for (n matrices)
together with the set of matrices of the form
- for ((n2−n)/2 matrices)
and the matrices
- for ((n2−n)/2 matrices)
where denotes the complex number , known as the imaginary unit.
If n orthonormal eigenvectors of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is where and therefore
where are the eigenvalues on the diagonal of the diagonal matrix .
Πρότυπο:AnchorAdditional facts related to Hermitian matrices include:
- The sum of a square matrix and its conjugate transpose is Hermitian.
- The difference of a square matrix and its conjugate transpose 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:
- The determinant of a Hermitian matrix is real:
- Therefore if
- (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
Frankel, Theodore (2004). The geometry of physics: an introduction. Cambridge University Press. σελ. 652. ISBN 0-521-53927-7. 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.
- ↑ Physics 125 Course Notes at California Institute of Technology
- Μαθηματική Μήτρα
- Ορθογώνια Μήτρα
- Μοναδιακή Μήτρα
- Hermitian form
- Self-adjoint operator
- Haynsworth inertia additivity formula
- Ομώνυμο άρθρο στην Βικιπαίδεια
- Ομώνυμο άρθρο στην Livepedia
- Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
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