Hermitian matrix

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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}} or A = \overline {A^\text{T}}, in matrix form.

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

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 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]

See the following example:


\begin{bmatrix}
2 & 2+i & 4 \\
2-i & 3 & i \\
4 & -i & 1 \\
\end{bmatrix}

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,[1][2] which results in skew-Hermitian matrices (see below).

Here we offer another useful Hermitian matrix using an abstract example. If a square matrix  A equals the multiplication of a matrix and its conjugate transpose, that is,  A=BB^\dagger , then  A is a Hermitian positive semi-definite matrix. Furthermore, if  B is row full-rank, then  A is positive definite.

Properties[edit]

  • The entries on the main diagonal (top left to bottom right) 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.
  • 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  v^\dagger A v is real because of  v^\dagger A v = (v^\dagger A v)^\dagger . 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 iIn 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:
\; 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 .

Further properties[edit]

Additional 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.)

Rayleigh quotient[edit]

Main article: Rayleigh quotient

See also[edit]

References[edit]

External links[edit]