Hermitian matrix
Hermitian-symmetric matrix, self-conjugate matrix
A square matrix over
that is the same as its Hermitian-conjugate matrix
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that is, a matrix whose entries satisfy the condition . If all the
, then a Hermitian matrix is symmetric (cf. Symmetric matrix). The Hermitian matrices of a fixed order form a vector space over
. If
and
are two Hermitian matrices of the same order, then so is
. Under the operation
the Hermitian matrices (of order
) form a Jordan algebra. The product
of two Hermitian matrices is itself Hermitian if and only if
and
commute.
The Hermitian matrices of order are the matrices of Hermitian transformations of an
-dimensional unitary space in an orthonormal basis (see Self-adjoint linear transformation). On the other hand, Hermitian matrices are the matrices of Hermitian forms in an
-dimensional complex vector space. Like Hermitian forms (cf. Hermitian form), Hermitian matrices can be defined over any skew-field with an anti-involution.
All eigen values of a Hermitian matrix are real. For every Hermitian matrix there exists a unitary matrix
such that
is a real diagonal matrix. A Hermitian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. Non-negative (positive-definite) Hermitian matrices correspond to non-negative (positive-definite) Hermitian linear transformations and Hermitian forms.
Comments
References
[a1] | F.R. [F.R. Gantmakher] Gantmacher, "Matrix theory" , 1–2 , Chelsea, reprint (1959) (Translated from Russian) |
[a2] | B. Noble, J.W. Daniel, "Applied linear algebra" , Prentice-Hall (1979) |
Hermitian matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_matrix&oldid=14435