Hermitian MatrixEdit
A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. In finite-dimensional linear algebra, this condition generalizes the familiar real symmetric matrices and endows the matrix with a range of especially well-behaved spectral properties. Hermitian matrices play a central role in many parts of mathematics, physics, and applied numerical methods, where they often model observable quantities, covariance-like structures, or stabilization properties in computations.
In many treatments, the distinction between Hermitian matrices and their real counterparts is one of degree rather than kind: every real symmetric matrix is Hermitian, but Hermitian matrices may have complex off-diagonal entries that come in conjugate pairs. The defining condition H = H* (where H* denotes the conjugate transpose of H) implies that diagonal entries are real and that aij = conjugate(aji) for all i, j. This restriction imposes strong consequences for eigenvalues, diagonalization, and the geometry of the associated quadratic forms.
Definition and basic facts
- A matrix H is Hermitian if H = H*, i.e., H equals its own conjugate transpose. This is the finite-dimensional incarnation of being self-adjoint in an inner product space.
- For a Hermitian matrix, all diagonal entries are real, and the off-diagonal entries satisfy aij = conjugate(aji).
- Hermitian matrices are a special case of normal matrices matrices, since H*H = HH*. They are automatically diagonalizable by a unitary matrix similarity transformation.
Key consequences of the Hermitian condition include: - Real eigenvalues: every eigenvalue of a Hermitian matrix is a real number, which is a reason these matrices model observables in physics and other real-valued measurements in mathematics. - Orthogonal eigenvectors: eigenvectors corresponding to distinct eigenvalues are orthogonal with respect to the standard inner product, and one can choose an orthonormal basis of eigenvectors. - Existence of a spectral decomposition: there exists a unitary matrix U such that U*HU is a real diagonal matrix D of eigenvalues, and equivalently H can be written as H = UDU*. This is the essence of the spectral theorem for Hermitian matrices. - Real trace and real determinant: the trace of a Hermitian matrix is real (equals the sum of its real eigenvalues), and the determinant is real (equals the product of the real eigenvalues).
Real symmetric matrices are exactly the Hermitian matrices with all entries real. Thus many results in real linear algebra extend to the complex setting with the Hermitian condition providing the necessary structure.
Spectral theorem and eigenstructure
The spectral theorem for Hermitian matrices states that any Hermitian matrix H can be unitarily diagonalized. Concretely, there exists a unitary matrix U such that U*HU = diag(λ1, λ2, ..., λn), where each λi is a real eigenvalue. The columns of U are an orthonormal set of eigenvectors of H. This theorem underpins many practical procedures: - Computing eigenvalues and eigenvectors in a stable way, since unitary transformations preserve norms. - Expressing quadratic forms x*Hx as a real-weighted sum of squares in an eigenbasis, which clarifies definiteness properties. - Stability analyses in applications where perturbations can be understood via contouring changes to eigenvalues.
Eigenvectors associated with distinct eigenvalues are orthogonal, and even in the presence of repeated eigenvalues, one can choose a complete set of orthonormal eigenvectors by an orthogonalization process within each eigenspace.
Positive definiteness and related notions
- Positive semidefinite: a Hermitian matrix H is positive semidefinite if x*Hx ≥ 0 for all vectors x. This is equivalent to all eigenvalues being nonnegative.
- Positive definite: H is positive definite if x*Hx > 0 for all nonzero x; equivalently, all eigenvalues are strictly positive.
- These notions play a crucial role in optimization, statistics (covariance-type structures), and numerical linear algebra (stability and conditioning).
The Gram matrix of a collection of vectors, G = X*X, is Hermitian and positive semidefinite. Such matrices frequently arise in data science and signal processing.
Extensions, distinctions, and terminology
In finite-dimensional spaces, the terms “Hermitian” and “self-adjoint” are usually interchangeable. In infinite-dimensional contexts, however, one must be careful: a linear operator may be symmetric (A ⊆ A*) but not necessarily self-adjoint (A* may properly extend A). In operator theory, the adjective “self-adjoint” is typically preferred for unbounded operators to emphasize the exact domain conditions required for equality with the adjoint.
From a computational perspective, special structures of Hermitian matrices can be exploited: - Efficient eigenvalue algorithms that preserve Hermitian structure, such as the Jacobi method or various unitary-QR variants, can be more stable and faster than general-purpose methods. - Positive definiteness can guide preconditioning and iterative methods for solving linear systems.
Applications of Hermitian matrices span physics, engineering, and data analysis. In quantum mechanics, observable quantities are represented by Hermitian operators on a Hilbert space, ensuring real measurement outcomes. In signal processing and communications, Hermitian matrices model certain correlation and covariance structures, while their spectral properties inform filter design and stability analyses.
Examples: - A 2×2 Hermitian matrix can be written as [[a, z], [conjugate(z), b]] with real a and b and complex z. Its eigenvalues are real, and it admits a unitary diagonalization. - A density matrix in quantum theory is Hermitian, positive semidefinite, and has trace one, encoding a statistical mixture of quantum states.