Hermitian Guide, Meaning , Facts, Information and Description
In this article we discuss the term hermitian as used in operator and matrix theory to refer to a certain kind of operator (or matrix). We warn the reader that there is a range of conflicting terminology in use in the literature generally and in Wikipedia in particular concerning these concepts. This problem is particularly acute when one compares works translated from other languages.
A number of mathematical entities are named hermitian, after the mathematician Charles Hermite.
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2 Elements of a C*-algebra 3 See also |
A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries so that the matrix is equal to its own conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:
In case the matrix has only real entries, a matrix is Hermitian if and only if it is symmetric with respect to the (top left to bottom right) diagonal of the matrix.
Every Hermitian matrix is normal, 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 means that all eigenvaluess of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. It is possible to find an orthonormal basis of Cn consisting only of eigenvectors.
If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite. Matrix theorists sometimes refer to real Hermitian matrices as symmetric matrices, since indeed they are symmetric with respect to the diagonal.
In a C*-algebra, we generalise this notion as discussed in self-adjoint.
This is an Article on Hermitian. Page Contains Information, Facts Details or Explanation Guide About Hermitian Matrices
For example,
is a Hermitian matrix.Elements of a C*-algebra
See also
For related concepts including symmetric and self-adjoint unbounded operators see Self-adjoint operator.