In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix 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:
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: A square matrix
is Hermitian if and only if it is equal to its conjugate transpose, that is, it satisfies
This is also the way that the more general concept of self-adjoint operator is defined.
is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.
Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues.
[2] The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information.
Hermitian matrices are extensively studied in linear algebra and numerical analysis.
The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.
Channel matrices in MIMO systems often exhibit Hermitian properties.
The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.
[4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.
The diagonal elements must be real, as they must be their own complex conjugate.
equals the product of a matrix with its conjugate transpose, that is,
The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.
This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors.
Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues.
Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of Cn consisting of n eigenvectors of A.
For an arbitrary complex valued vector v the product
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, ℂ, 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.
If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:
is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of
Additional facts related to Hermitian matrices include: In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[10]
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose
It can be shown[9] that, for a given matrix, the Rayleigh quotient reaches its minimum value
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues.
Specifically, this is the basis for Rayleigh quotient iteration.
When the matrix is Hermitian, the numerical range is equal to the spectral norm.