Hermitian matrix

Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: Equality with the adjoint A square matrix A is Hermitian if and only if it is equal to its conjugate transpose, that is, it satisfies \langle \mathbf w, A \mathbf v\rangle = \langle A \mathbf w, \mathbf v\rangle, for any pair of vectors \mathbf v, \mathbf w, where \langle \cdot, \cdot\rangle denotes 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 n\times{}n matrix A is Hermitian if and only if \langle \mathbf{v}, A \mathbf{v}\rangle\in\R, \quad \text{for all } \mathbf{v}\in \mathbb{C}^{n}. Spectral properties A square matrix A 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 a of an operator \hat{A} on some quantum state |\psi\rangle 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 (SVD) 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 (MIMO) 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 A is denoted as A^\mathsf{H} , the transpose of matrix A is denoted as A^\mathsf{T} and conjugate of matrix A is denoted as \overline{A} . See the following example: \begin{bmatrix} 0 & a - ib & c-id \\ a+ib & 1 & m-in \\ c+id & m+in & 2 \end{bmatrix} 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 A equals the product of a matrix with its conjugate transpose, that is, A = BB^\mathsf{H} , then A is a Hermitian positive semi-definite matrix. Furthermore, if B is row full-rank, then A is positive definite. ==Properties==

Main diagonal values are real The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real. {{math proof|1= By definition of the Hermitian matrix H_{ij} = \overline{H}_{ji} so for the above follows, as a number can equal its complex conjugate only if the imaginary parts are zero. }} 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. {{math proof|1= H_{ij} = \overline{H}_{ji} by definition. Thus H_{ij} = H_{ji} (matrix symmetry) if and only if H_{ij} = \overline{H}_{ij} (H_{ij} is real). }} So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit i, then it becomes Hermitian. Normal Every Hermitian matrix is a normal matrix. That is to say, AA^\mathsf{H} = A^\mathsf{H}A. {{math proof|1=A = A^\mathsf{H}, so AA^\mathsf{H} = AA = A^\mathsf{H}A.}} 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. {{math proof|1= (A + B)_{ij} = A_{ij} + B_{ij} = \overline{A}_{ji} + \overline{B}_{ji} = \overline{(A + B)}_{ji}, as claimed.}} Inverse is Hermitian The inverse of an invertible Hermitian matrix is Hermitian as well. {{math proof|1= If A^{-1}A = I, then I= I^\mathsf{H} = \left(A^{-1}A\right)^\mathsf{H} = A^\mathsf{H}\left(A^{-1}\right)^\mathsf{H} = A \left(A^{-1}\right)^\mathsf{H}, so A^{-1}=\left(A^{-1}\right)^\mathsf{H} as claimed.}} Associative product of Hermitian matrices The product of two Hermitian matrices and is Hermitian if and only if . {{math proof|1= (AB)^\mathsf{H} = \overline{(AB)^\mathsf{T}} = \overline{B^\mathsf{T} A^\mathsf{T}} = \overline{B^\mathsf{T}} \ \overline{A^\mathsf{T}} = B^\mathsf{H} A^\mathsf{H} = BA. Thus (AB)^\mathsf{H} = AB if and only if AB = BA. Thus is Hermitian if is Hermitian and is an integer. }} ABA Hermitian If A and B are Hermitian, then ABA is also Hermitian. {{math proof|1= (ABA)^\mathsf{H} = (A(BA))^\mathsf{H} = (BA)^\mathsf{H}A^\mathsf{H} = A^\mathsf{H}B^\mathsf{H}A^\mathsf{H} = ABA }} is real for complex For an arbitrary complex valued vector the product \mathbf{v}^\mathsf{H} A \mathbf{v} is real because of \mathbf{v}^\mathsf{H} A \mathbf{v} = \left(\mathbf{v}^\mathsf{H} A \mathbf{v}\right)^\mathsf{H} . 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 (orthonormal with respect to the Frobenius inner product) can be described as follows: E_{jj} \text{ for } 1 \leq j \leq n \quad (n \text{ matrices}) together with the set of matrices of the form \frac{1}{\sqrt{2}}\left(E_{jk} + E_{kj}\right) \text{ for } 1 \leq j and the matrices \frac{i}{\sqrt{2}}\left(E_{jk} - E_{kj}\right) \text{ for } 1 \leq j where i denotes the imaginary unit, i = \sqrt{-1}~. 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 \mathbf{u}_1, \dots, \mathbf{u}_n of a Hermitian matrix are chosen and written as the columns of the matrix , then one eigendecomposition of is A = U \Lambda U^\mathsf{H} where U U^\mathsf{H} = I = U^\mathsf{H} U and therefore A = \sum_j \lambda_j \mathbf{u}_j \mathbf{u}_j ^\mathsf{H}, where \lambda_j are the eigenvalues on the diagonal of the diagonal matrix \Lambda. Singular values The singular values of A are the absolute values of its eigenvalues: Since A has an eigendecomposition A=U\Lambda U^H, where U is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of A is A=U|\Lambda|\text{sgn}(\Lambda)U^H, where |\Lambda| and \text{sgn}(\Lambda) are diagonal matrices containing the absolute values |\lambda| and signs \text{sgn}(\lambda) of A's eigenvalues, respectively. \sgn(\Lambda)U^H is unitary, since the columns of U^H are only getting multiplied by \pm 1. |\Lambda| contains the singular values of A, namely, the absolute values of its eigenvalues. Real determinant The determinant of a Hermitian matrix is real: {{math proof|1= \det(A) = \det\left(A^\mathsf{T}\right)\quad \Rightarrow \quad \det\left(A^\mathsf{H}\right) = \overline{\det(A)} Therefore if A = A^\mathsf{H}\quad \Rightarrow \quad \det(A) = \overline{\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.) ==Decomposition into Hermitian and skew-Hermitian matrices==

Additional facts related to Hermitian matrices include: • The sum of a square matrix and its conjugate transpose \left(A + A^\mathsf{H}\right) is Hermitian. • The difference of a square matrix and its conjugate transpose \left(A - A^\mathsf{H}\right) is skew-Hermitian (also called antihermitian). 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 . C = A + B \quad\text{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\text{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right) ==Rayleigh quotient==

In mathematics, for a given complex Hermitian matrix and nonzero vector , the Rayleigh quotient R(M, \mathbf{x}), is defined as: R(M, \mathbf{x}) := \frac{\mathbf{x}^\mathsf{H} M \mathbf{x}}{\mathbf{x}^\mathsf{H} \mathbf{x}}. For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose \mathbf{x}^\mathsf{H} to the usual transpose \mathbf{x}^\mathsf{T}. R(M, c \mathbf x) = R(M, \mathbf x) for any non-zero real scalar c. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value \lambda_\min (the smallest eigenvalue of M) when \mathbf x is \mathbf v_\min (the corresponding eigenvector). Similarly, R(M, \mathbf x) \leq \lambda_\max and R(M, \mathbf v_\max) = \lambda_\max . 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 (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, \lambda_\max 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. ==See also==