In mathematics, a Hermitian 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: or , in matrix form.
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix is denoted by , then the Hermitian property can
be written concisely as
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 See the following example:
The diagonal elements must be real, as they must be their own complex conjugate.
Well-known families of Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian.
In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,
which results in skew-Hermitian matrices. Here we offer another useful Hermitian matrix
using an abstract example. If a square matrix equals the multiplication of a matrix and
its conjugate transpose, that is, , then is a Hermitian matrix. Furthermore, if is column
full-rank, then is positive semi-definite; else if is row full-rank, then is positive
definite. Properties
The entries on the main diagonal of any Hermitian matrix are necessarily real, because they
have to be equal to their complex conjugate. 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. The finite-dimensional spectral theorem 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 and only if AB = BA. Thus An is Hermitian if A is Hermitian and n is
an integer. For an arbitrary complex valued vector v the
product is real because of . This is especially important in quantum physics where hermitian
matrices are operators that measure properties of a system e.g. total spin which have to
be real. The Hermitian complex n-by-n matrices do not
form a vector space over the complex numbers, since the identity matrix In is Hermitian,
but i In is not. However the complex Hermitian matrices do form a vector space over the real
numbers R. In the 2n2-dimensional vector space of complex n × n matrices over R, 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
together with the set of matrices of the form for
and the matrices for
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 .
Further properties Additional 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. 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:
Proof: 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.)
Rayleigh quotient
See also Skew-Hermitian matrix
Haynsworth inertia additivity formula Hermitian form
Self-adjoint operator Unitary matrix
References
External links Hazewinkel, Michiel, ed., "Hermitian matrix",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Visualizing Hermitian Matrix as An Ellipse
with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
"Hermitian Matrices" at MathPages.com.