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...

square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That...

x . {\displaystyle \mathbf {x} .} More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite...

Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle \mathbf {A} } is an n × m {\displaystyle n\times m} matrix...

overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if A T = A ¯ . {\displaystyle...

matrix (or Gramian matrix, Gramian) of a set of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} in an inner product space is the Hermitian matrix...

contained in R {\displaystyle \mathbb {R} } . In other words: A normal matrix is Hermitian if and only if all its eigenvalues are real. In general, the sum...

space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose...

(right/left). Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0 ), the Pauli matrices...

of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form...

the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted...

orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional...

a Hermitian form ( V , h ) {\displaystyle (V,h)} is called a Hermitian space. The matrix representation of a complex Hermitian form is a Hermitian matrix...

represent cross products as matrix multiplications. Furthermore, if A {\displaystyle A} is a skew-symmetric (or skew-Hermitian) matrix, then x T A x = 0 {\displaystyle...

to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix. Let A , B {\textstyle...

quotient of the eigenvector). In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a product...

criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if...

covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the...

real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary...

special case where ⁠ A {\displaystyle A} ⁠ is a normal matrix (for example, a Hermitian matrix), the pseudoinverse ⁠ A + {\displaystyle A^{+}} ⁠ annihilates...

When a complex matrix A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} is a Hermitian matrix (or more generally a normal matrix), eigenvectors...

symmetric matrix and more generally a Hermitian matrix, and a unitary matrix, is never defective; more generally, a normal matrix (which includes Hermitian and...

mathematics, the Rayleigh quotient (/ˈreɪ.li/) for a given complex Hermitian matrix M {\displaystyle M} and nonzero vector x {\displaystyle x} is defined...

apply, with symmetric matrix, quadratic form, bilinear form, and transpose xT replaced respectively by Hermitian matrix, Hermitian form, sesquilinear form...

vector bundle Hermitian matrix, a square matrix with complex entries that is equal to its own conjugate transpose Skew-Hermitian matrix Hermitian operator...

Density matrix — a matrix describing the statistical state of a quantum system. Hermitian, non-negative and with trace 1. Fundamental matrix (computer...

adjacency matrix are complex, but the Kirchhoff matrix remains symmetric, rather than being Hermitian. Such a matrix is usually called an "admittance matrix",...

Another way of saying this is that a unitary matrix is the exponential of i times a Hermitian matrix, so that the additive conserved real quantity,...

after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial...

important example. This includes the symmetric or Hermitian Hessenberg matrices. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices...