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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: is Hermitian {\displaystyle A {\text { is ...
Hermitian manifold. In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a ...
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space[1][2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive ...
Rayleigh quotient. In mathematics, the Rayleigh quotient[1] (/ ˈreɪ.li /) for a given complex Hermitian matrix and nonzero vector is defined as: [2][3] For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose to the usual transpose . Note that for any non-zero scalar .
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. [1] The special case of a 1-dimensional algebra is known as a dual number.
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z.Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R n, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only.
Weyl's inequality. In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.