Hermitian matrix
In mathematics, a Hermitian matrix is a square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and. In index form, 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 always having real eigenvalues. Other, equivalent notations in common use are although in quantum mechanics, typically means the complex conjugate only, and not the conjugate transpose.
Alternative characterizations
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:Equality with the adjoint
A square matrix is Hermitian if and only if it is equal to its conjugate transpose, that is, it satisfiesfor any pair of vectors where denotes product space|the inner product] operation.
This is also the way that the more general concept of self-adjoint operator is defined.
Real-valuedness of quadratic forms
An matrix is Hermitian if and only ifSpectral properties
A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.Applications
Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue of an operator on some quantum state is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.In signal processing, Hermitian matrices are utilized in tasks like Fourier analysis and signal representation. 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. They have well-defined spectral properties, and many numerical algorithms, such as the Lanczos algorithm, exploit these properties for efficient computations. Hermitian matrices also appear in techniques like singular value decomposition and eigenvalue decomposition.
In statistics and machine learning, Hermitian matrices are used in covariance matrices, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.
Hermitian matrices are applied in the design and analysis of communications system, especially in the field of multiple-input multiple-output systems. Channel matrices in MIMO systems often exhibit Hermitian properties.
In graph theory, Hermitian matrices are used to study the spectra of graphs. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. 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.
Examples and solutions
In this section, the conjugate transpose of matrix is denoted as the transpose of matrix is denoted as and conjugate of matrix is denoted asSee the following example:
The diagonal elements must be real, as they must be their own complex conjugate.
Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. 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 product of a matrix with its conjugate transpose, that is, then is a Hermitian positive semi-definite matrix. Furthermore, if is row full-rank, then is positive definite.
Properties
Main diagonal values are real
The entries on the main diagonal of any Hermitian matrix are real.Only the main diagonal entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their off-diagonal elements, as long as diagonally-opposite entries are complex conjugates.
Symmetric
A matrix that has only real entries is symmetric if and only if it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit then it becomes Hermitian.
Normal
Every Hermitian matrix is a normal matrix. That is to say,Diagonalizable
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 with dimension are real, and that has 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 consisting of eigenvectors of.Sum of Hermitian matrices
The sum of any two Hermitian matrices is Hermitian.Inverse is Hermitian
The inverse of an invertible Hermitian matrix is Hermitian as well.Associative product of Hermitian matrices
The product of two Hermitian matrices and is Hermitian if and only if.''ABA'' Hermitian
If A and B are Hermitian, then ABA is also Hermitian.is real for complex
For an arbitrary complex valued vector 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.Complex Hermitian forms vector space over
The Hermitian complex -by- matrices do not form a vector space over the complex numbers,, since the identity matrix is Hermitian, but is not. However the complex Hermitian matrices do form a vector space over the real numbers. In the -dimensional vector space of complex matrices over, the complex Hermitian matrices form a subspace of dimension. If denotes the -by- matrix with a in the position and zeros elsewhere, a basis can be described as follows:together with the set of matrices of the form
and the matrices
where denotes the imaginary unit,
An example is that the four Pauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over.
Eigendecomposition
If orthonormal eigenvectors of a Hermitian matrix are chosen and written as the columns of the matrix, then one eigendecomposition of is where and thereforewhere are the eigenvalues on the diagonal of the diagonal matrix
Singular values
The singular values of are the absolute values of its eigenvalues:Since has an eigendecomposition, where is a unitary matrix, a singular value decomposition of is, where and are diagonal matrices containing the absolute values and signs of 's eigenvalues, respectively. is unitary, since the columns of are only getting multiplied by. contains the singular values of, namely, the absolute values of its eigenvalues.
Real determinant
The determinant of a Hermitian matrix is real:Decomposition into Hermitian and skew-Hermitian matrices
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 the commutator of two Hermitian matrices is skew-Hermitian.
- An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix. This is known as the Toeplitz decomposition of.
Rayleigh quotient
In mathematics, for a given complex Hermitian matrix and nonzero vector, the Rayleigh quotient is defined as:For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose for any non-zero real scalar Also, recall that a Hermitian matrix has real eigenvalues.
It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value when is . Similarly, and
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
The range of the Rayleigh quotient is called a numerical range. When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.