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The operator is said to be positive-definite, and written >, if , >, for all {}. [ 1 ] Many authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator.
Hermitian form, a specific sesquilinear form; Hermitian function, a complex function whose complex conjugate is equal to the original function with the variable changed in sign; Hermitian manifold/structure Hermitian metric, is a smoothly varying positive-definite Hermitian form on each fiber of a complex vector bundle
(non-Unicode name) ('Scarab' is an informal name for the generic currency sign) § Section sign: section symbol, section mark, double-s, 'silcrow' Pilcrow; Semicolon: Colon ℠ Service mark symbol: Trademark symbol / Slash (non-Unicode name) Division sign, Forward Slash: also known as "stroke" / Solidus (the most common of the slash symbols ...
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Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint).
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré . He was the first to prove that e , the base of natural logarithms , is a transcendental number .
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